Zhouli Xu (Joint with Michael Hopkins, Jianfeng Lin, and ... · The geography problem of...

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

I N: closed simply connected topological 4-manifold.

I Two important invariants of N:

QN : H2(N;Z)× H2(N;Z) −→ Z,(a, b) 7−→ 〈a ∪ b, [N]〉.

Question

QN : H2(N;Z)× H2(N;Z) −→ Z,(a, b) 7−→ 〈a ∪ b, [N]〉.

Question

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]〉.

Question

QN : H2(N;Z)× H2(N;Z) −→ Z,(a, b) 7−→ 〈a ∪ b, [N]〉.

Question

QN : H2(N;Z)× H2(N;Z) −→ Z,(a, b) 7−→ 〈a ∪ b, [N]〉.

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)

8 mod 2)

can be realized

Theorem (Freedman)

8 mod 2)

can be realized

Theorem (Freedman)

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

Theorem

Smooth category

Question

Theorem

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

The Geography Problem

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

Theorem (Donaldson)

Q: definite

Theorem (Donaldson)

Q: 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

2. Q: even

Q ∼= kE8 ⊕ q

(0 11 0

Indefinite forms

indefinite

not even even

2. Q: even

Q ∼= kE8 ⊕ q

(0 11 0

indefinite

not even even

Q: not evenQ can be realized by a connected sum of copies of CP2 and CP2

indefinite

not even even

Q: not evenQ can be realized by a connected sum of copies of CP2 and CP2

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

definite indefinite

not even even

I Q ∼= kE8 ⊕ q

(0 11 0

), k ∈ Z, q ∈ N

definite indefinite

not even even

I Q ∼= kE8 ⊕ q

(0 11 0

), k ∈ Z, q ∈ N

definite indefinite

not even even

I Q ∼= kE8 ⊕ q

(0 11 0

), k ∈ Z, q ∈ N

definite indefinite

not even even

I Q ∼= kE8 ⊕ q

(0 11 0

), k ∈ Z, q ∈ N

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 form

2pE8 ⊕ q

(0 11 0

#pK3 #

q−3p(S2 × S2)

I K3: 2E8 ⊕ 3

(0 11 0

)I S2 × S2:

(0 11 0

The 118 -Conjecture

The form

2pE8 ⊕ q

(0 11 0

#pK3 #

q−3p(S2 × S2)

I K3: 2E8 ⊕ 3

(0 11 0

)I S2 × S2:

(0 11 0

The 118 -Conjecture

The form

2pE8 ⊕ q

(0 11 0

#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:

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

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.

Definition

Q: evenQ is spin realizable if it can be realized by a closed smooth spin4-manifold.

Theorem (Furuta)

2pE8 ⊕ q

(0 11 0

Definition

Q: evenQ is spin realizable if it can be realized by a closed smooth spin4-manifold.

Theorem (Furuta)

2pE8 ⊕ q

(0 11 0

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.

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,

2p + 2 p ≡ 1, 2, 5, 6 (mod 8)

2p + 3 p ≡ 3, 4, 7 (mod 8)

2p + 4 p ≡ 0 (mod 8).

(0 11 0

2p + 2 p ≡ 1, 2, 5, 6 (mod 8)

2p + 3 p ≡ 3, 4, 7 (mod 8)

2p + 4 p ≡ 0 (mod 8).

(0 11 0

2p + 2 p ≡ 1, 2, 5, 6 (mod 8)

2p + 3 p ≡ 3, 4, 7 (mod 8)

2p + 4 p ≡ 0 (mod 8).

(0 11 0

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

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)

Dφ+ ρ(a)φ = 0d+a− ρ−1(φφ∗)0 = 0

d∗a = 0

Dφ+ ρ(a)φ = 0d+a− ρ−1(φφ∗)0 = 0

d∗a = 0

Dφ+ ρ(a)φ = 0d+a− ρ−1(φφ∗)0 = 0

d∗a = 0

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)

I SH1 = H1/(H1 \ B(H1,R))

I SH2 = H2/(H2 \ B(H2, ε))

I SW induces a Pin(2)-equivariant map between spheres

: SH1 −→ SH2

Furuta’s idea

1. SW (0) = 0

I SH1 = H1/(H1 \ B(H1,R))

I SH2 = H2/(H2 \ B(H2, ε))

: SH1 −→ SH2

Furuta’s idea

1. SW (0) = 0

I SH1 = H1/(H1 \ B(H1,R))

I SH2 = H2/(H2 \ B(H2, ε))

: SH1 −→ SH2

Furuta’s idea

1. SW (0) = 0

I SH1 = H1/(H1 \ B(H1,R))

I SH2 = H2/(H2 \ B(H2, ε))

: SH1 −→ SH2

Furuta’s idea

1. SW (0) = 0

I SH1 = H1/(H1 \ B(H1,R))

I SH2 = H2/(H2 \ B(H2, ε))

: SH1 −→ SH2

Furuta’s idea

1. SW (0) = 0

I SH1 = H1/(H1 \ B(H1,R))

I SH2 = H2/(H2 \ B(H2, ε))

: SH1 −→ SH2

I Problem: SH1 and SH2 are both infinite dimensional

I In order to use homotopy theory, we want maps between finitedimensional spheres

: SH1 −→ SH2

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

I SW = L + CI L: linear Fredholm operator

I C : nonlinear operatorbounded sets 7−→ compact sets

I V1 = L−1(V2)

I SW apr := L + PrV2 C : V1 −→ V2

I SW = L + CI L: linear Fredholm operatorI C : nonlinear operator

I V1 = L−1(V2)

I SW apr := L + PrV2 C : V1 −→ V2

I V1 = L−1(V2)

I SW apr := L + PrV2 C : V1 −→ V2

I V1 = L−1(V2)

I SW apr := L + PrV2 C : V1 −→ V2

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 apr

1. SW apr(0) = 0

SW apr

1. SW apr(0) = 0

2. SW apr is a Pin(2)-equivariant map

3. When V2 is large enough,

SW apr

1. SW apr(0) = 0

SW apr

1. SW apr(0) = 0

SW apr

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

apr : SV1 −→ SV2

SW apr

I SV1 = B(V1,R + 1)/S(V1,R + 1)

I SV2 = V2/(V2 \ B(V2, ε))

SW apr

I SV1 = B(V1,R + 1)/S(V1,R + 1)

I SV2 = V2/(V2 \ B(V2, ε))

SW apr

I SV1 = B(V1,R + 1)/S(V1,R + 1)

I SV2 = V2/(V2 \ B(V2, ε))

S0 SV2

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(∞) =∞

S0 SV2

Pin(2) Z/2

apr(0) = 0, SW+

apr(∞) =∞

S0 SV2

I H: 4-dimensional, Pin(2) acts via left multiplication

I R: 1-dimensional, pull back of the sign representation viaPin(2) Z/2

apr(0) = 0, SW+

apr(∞) =∞

S0 SV2

Pin(2) Z/2

apr(0) = 0, SW+

apr(∞) =∞

S0 SV2

Pin(2) Z/2

I Pin(2)-fixed points of SV1 and SV2 are both S0 = 0 ∪ ∞

apr(0) = 0, SW+

apr(∞) =∞

S0 SV2

Pin(2) Z/2

apr(0) = 0, SW+

apr(∞) =∞

S0 SV2

Pin(2) Z/2

apr(0) = 0, SW+

apr(∞) =∞

S0 SV2

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)

S0 SV2

apr =⇒SpH

S0 SqR

BF (M)

(0 11 0

thenV1 − V2

∼= pH− qR

S0 SV2

apr =⇒SpH

S0 SqR

BF (M)

(0 11 0

thenV1 − V2

∼= pH− qR

S0 SV2

apr =⇒SpH

S0 SqR

BF (M)

(0 11 0

thenV1 − V2

∼= pH− qR

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

S0 SqR

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

Jones’ conjecture

Conjecture (Jones)

For p ≥ 2, a level-(p, q) Furuta–Mahowald class exists if and onlyif

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)

2p + 2 p ≡ 1 (mod 4)

2p + 2 p ≡ 2 (mod 4)

2p + 3 p ≡ 3 (mod 4)

2p + 4 p ≡ 0 (mod 4).

Jones’ conjecture

Conjecture (Jones)

2p + 2 p ≡ 1 (mod 4)

2p + 2 p ≡ 2 (mod 4)

2p + 3 p ≡ 3 (mod 4)

2p + 4 p ≡ 0 (mod 4).

Theorem (Furuta–Kamitani)

For p ≥ 2, a level-(p, q) Furuta–Mahowald class exists only if

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

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

Main Theorem

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)

b2(M) ≥ 10

8| sign(M)|+ 4.

In the sense of classifying all Furuta–Mahowald classes oflevel-(p, q), this is the limit

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

S0 SqR

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

0 = [SV , S0]G

I Equivariantly, there are more spheres!

V : G -representation, πGV S0 = [SV ,S0]G

0 = [SV , S0]G

Non-nilpotent elements in πGFS

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

1. p : S0 −→ S0

−VS0

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

−VS0

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

−VS0

1. p : S0 −→ S0

−VS0

1. p : S0 −→ S0

−VS0

1. p : S0 −→ S0

I V : real nontrivial irreducible representation

I stable class in πG−VS

1. p : S0 −→ S0

−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

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|

S−(k+1)|β|

S0 S−|α|

@γ′

I α, β ∈ πGFS0

Definition

I |MGβ (α)| = |γ| = |α| − k|β|

S−k|β|

S0 S−|α|

S−(k+1)|β|

S0 S−|α|

@γ′

I α, β ∈ πGFS0

Definition

I |MGβ (α)| = |γ| = |α| − k|β|

S−k|β|

S0 S−|α|

S−(k+1)|β|

S0 S−|α|

@γ′

I α, β ∈ πGFS0

Definition

I |MGβ (α)| = |γ| = |α| − k|β|

S−k|β|

S0 S−|α|

S−(k+1)|β|

S0 S−|α|

@γ′

I α, β ∈ πGFS0

Definition

I |MGβ (α)| = |γ| = |α| − k|β|

S−k|β|

S0 S−|α|

S−(k+1)|β|

S0 S−|α|

@γ′

I α, β ∈ πGFS0

Definition

I |MGβ (α)| = |γ| = |α| − k|β|

S−k|β|

S0 S−|α|

S−(k+1)|β|

S0 S−|α|

@γ′

S−(k+1)|β|

S−k|β|

S−2|β|

S−|β|

S0 S−|α|

=⇒ |MGβ (α)| − |α| = −k|β|

S−(k+1)|β|

S−k|β|

S−2|β|

S−|β|

S0 S−|α|

=⇒ |MGβ (α)| − |α| = −k|β|

S−(k+1)|β|

S−k|β|

S−2|β|

S−|β|

S0 S−|α|

=⇒ |MGβ (α)| − |α| = −k|β|

S−(k+1)|β|

S−k|β|

S−2|β|

S−|β|

S0 S−|α|

=⇒ |MGβ (α)| − |α| = −k|β|

S−(k+1)|β|

S−k|β|

S−2|β|

S−|β|

S0 S−|α|

=⇒ |MGβ (α)| − |α| = −k|β|

S−(k+1)|β|

S−k|β|

S−2|β|

S−|β|

S0 S−|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.

S0 Skσ

S (k+1)σ

S0 Skσ

@γ′

ak+1σ

I G = C2, cyclic group of order 2I RO(C2) = Z⊕ Z,

generated by 1 and σI 1: trivial representationI σ: sign representation

S0 Skσ

S (k+1)σ

S0 Skσ

@γ′

ak+1σ

I G = C2, cyclic group of order 2I RO(C2) = Z⊕ Z, generated by 1 and σ

I 1: trivial representationI σ: sign representation

S0 Skσ

S (k+1)σ

S0 Skσ

@γ′

ak+1σ

I 1: trivial representation

I σ: sign representation

S0 Skσ

S (k+1)σ

S0 Skσ

@γ′

ak+1σ

S0 Skσ

S (k+1)σ

S0 Skσ

@γ′

ak+1σ

S0 Skσ

S (k+1)σ

S0 Skσ

@γ′

ak+1σ

S0 Skσ

S (k+1)σ

S0 Skσ

@γ′

ak+1σ

S0 Skσ

S (k+1)σ

S0 Skσ

@γ′

ak+1σ

S0 Skσ

S (k+1)σ

S0 Skσ

@γ′

ak+1σ

Classical Mahowald invariant

Sn+kσ Sn+k

Sn S0 S0

M(α)akσ

(ΦC2 )−1α

forget

• 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)

Sn+kσ Sn+k

Sn S0 S0

M(α)akσ

(ΦC2 )−1α

forget

• α ∈ πnS0

Sn+kσ Sn+k

Sn S0 S0

M(α)akσ

(ΦC2 )−1α

forget

• α ∈ πnS0

Sn+kσ Sn+k

Sn S0 S0

M(α)akσ

(ΦC2 )−1α

forget

• α ∈ πnS0

Sn+kσ Sn+k

Sn S0 S0

M(α)akσ

(ΦC2 )−1α

forget

• α ∈ πnS0

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.

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

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 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 1: trivial representation

I σ: sign representationI λ: 2-dimensional, rotation by 90

I λ: 2-dimensional, rotation by 90

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

|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.

|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.

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

I G = Pin(2)

I Irreducible representations H and R

I By definition, a level-(p, q) Furuta–Mahowald class exists

R)|+ qR is ≥ p

I G = Pin(2)

R)|+ qR is ≥ p

I G = Pin(2)

R)|+ qR is ≥ p

Main Theorem

|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

|MPin(2)aH (aq

R)|+ qR

I Had it been (8k + 3)H instead, our result would be 8-periodic

I Jones’ conjecture would be true

Main Theorem

|MPin(2)aH (aq

R)|+ qR

Main Theorem

|MPin(2)aH (aq

R)|+ qR

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).

=⇒ X (m + 1) −→ X (m)=⇒ X (m + 1) −→ X (m) −→ Σ−mCP∞

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∞

=⇒ X (m + 1) −→ X (m)

=⇒ X (m + 1) −→ X (m) −→ Σ−mCP∞

=⇒ X (m + 1) −→ X (m)=⇒ X (m + 1) −→ X (m) −→ Σ−mCP∞

The Mahowald Line

Consider the diagram

S0 SqR

S(pH)+

∃apHaqR

The Mahowald Line

S0 SqR

S(pH)+

∃apHaqR

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

S0 SqR

S(pH)+

∃apHaqR

I g is zero ⇐⇒

S−qR ∧ S(pH)+ → S(pH)+f−→ S0 is zero

The Mahowald Line

S0 SqR

S(pH)+

∃apHaqR

f−→ S0 is zero

The Mahowald Line

S0 SqR

S(pH)+

∃apHaqR

f−→ S0 is zero

The Mahowald Line

S0 SqR

S(pH)+

∃apHaqR

f−→ S0 is zero

The Mahowald Line

S0 SqR

S(pH)+

∃apHaqR

f−→ 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

=(S−qσ ∧ CP2p−1

= (4p − 2− q)-skeleton of X (q)

S0 SqR

S(pH)+

∃apHaqR

⇐⇒ X (q)4p−2−q −→ S0 is zero

The Mahowald Line

(S−qR ∧ S(pH)+)h Pin(2) =

((S−qR ∧ S(pH)+)hS1

S0 SqR

S(pH)+

∃apHaqR

⇐⇒ X (q)4p−2−q −→ S0 is zero

The Mahowald Line

(S−qR ∧ S(pH)+)h Pin(2) =(

S0 SqR

S(pH)+

∃apHaqR

⇐⇒ X (q)4p−2−q −→ S0 is zero

The Mahowald Line

(S−qR ∧ S(pH)+)h Pin(2) =(

S0 SqR

S(pH)+

∃apHaqR

⇐⇒ X (q)4p−2−q −→ S0 is zero

The Mahowald Line

(S−qR ∧ S(pH)+)h Pin(2) =(

S0 SqR

S(pH)+

∃apHaqR

⇐⇒ X (q)4p−2−q −→ S0 is zero

The Mahowald Line

(S−qR ∧ S(pH)+)h Pin(2) =(

S0 SqR

S(pH)+

∃apHaqR

⇐⇒ X (q)4p−2−q −→ S0 is zero

The Mahowald Line

(S−qR ∧ S(pH)+)h Pin(2) =(

S0 SqR

S(pH)+

∃apHaqR

⇐⇒ X (q)4p−2−q −→ S0 is zero

S2H ∃−→ S8R

S3H @−→ S8R

Lower bound

Classical Adams spectral sequence

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

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

h0 h1 h2 h3

Ph1 Ph2

d0 h30h4

P 3h1 P 3h2

h100 h5

h0h2h5

h20h3h5

h1h3h5

P 2h20i

h70Q′

P 4h20i

E1 + C0

X2 + C′

h250 h6

h2A′

B5 + D′2

h3A′

h3(E1 + C0)

p1 + h20h3h6p1 + h20h3h6

h1h3H1

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

π8 · η2 = 0π12 = 0π13 = 0

Now we start the induction

Inductive hypothesis

π8 · η2 = 0π12 = 0π13 = 0

Induction finished!

Intuition for a technical step

Another mini-movie

π4 = 0

π5 = 0

η · π6 = 0

π4 = 0

π5 = 0

η · π6 = 0

Zhouli Xu (Joint with Michael Hopkins, Jianfeng Lin, and ... · The geography problem of...

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