The Slice Spectral Sequence ofa Height 4 Theory

XiaoLin Danny Shi

(Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)

Harvard

June 8, 2018

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1

Atiyah’s Real K-theory KR

I C2: cyclic group of order 2 with generator τ

I Real space X : X with a C2-action

Xτ //

id

77Xτ // X

I Real vector bundle E over X :I E : complex vector bundle over XI E : Real spaceI p : E → X is C2-equivariantI τ : Ex → Eτ(x) is anti C-linear

τ(z · v) = z · τ(v)

(This is NOT a C2-equivariant complex vector bundle!)

I KR(X ): Grothendieck’s construction

Atiyah’s Real K-theory KR

I C2: cyclic group of order 2 with generator τ

I Real space X : X with a C2-action

Xτ //

id

77Xτ // X

I Real vector bundle E over X :I E : complex vector bundle over XI E : Real spaceI p : E → X is C2-equivariantI τ : Ex → Eτ(x) is anti C-linear

τ(z · v) = z · τ(v)

(This is NOT a C2-equivariant complex vector bundle!)

I KR(X ): Grothendieck’s construction

Atiyah’s Real K-theory KR

I C2: cyclic group of order 2 with generator τ

I Real space X : X with a C2-action

Xτ //

id

77Xτ // X

I Real vector bundle E over X :I E : complex vector bundle over XI E : Real spaceI p : E → X is C2-equivariantI τ : Ex → Eτ(x) is anti C-linear

τ(z · v) = z · τ(v)

(This is NOT a C2-equivariant complex vector bundle!)

I KR(X ): Grothendieck’s construction

Atiyah’s Real K-theory KR

I C2: cyclic group of order 2 with generator τ

I Real space X : X with a C2-action

Xτ //

id

77Xτ // X

I Real vector bundle E over X :

I E : complex vector bundle over XI E : Real spaceI p : E → X is C2-equivariantI τ : Ex → Eτ(x) is anti C-linear

τ(z · v) = z · τ(v)

(This is NOT a C2-equivariant complex vector bundle!)

I KR(X ): Grothendieck’s construction

Atiyah’s Real K-theory KR

I C2: cyclic group of order 2 with generator τ

I Real space X : X with a C2-action

Xτ //

id

77Xτ // X

I Real vector bundle E over X :I E : complex vector bundle over X

I E : Real spaceI p : E → X is C2-equivariantI τ : Ex → Eτ(x) is anti C-linear

τ(z · v) = z · τ(v)

(This is NOT a C2-equivariant complex vector bundle!)

I KR(X ): Grothendieck’s construction

Atiyah’s Real K-theory KR

I C2: cyclic group of order 2 with generator τ

I Real space X : X with a C2-action

Xτ //

id

77Xτ // X

I Real vector bundle E over X :I E : complex vector bundle over XI E : Real spaceI p : E → X is C2-equivariant

I τ : Ex → Eτ(x) is anti C-linear

τ(z · v) = z · τ(v)

(This is NOT a C2-equivariant complex vector bundle!)

I KR(X ): Grothendieck’s construction

Atiyah’s Real K-theory KR

I C2: cyclic group of order 2 with generator τ

I Real space X : X with a C2-action

Xτ //

id

77Xτ // X

I Real vector bundle E over X :I E : complex vector bundle over XI E : Real spaceI p : E → X is C2-equivariantI τ : Ex → Eτ(x) is anti C-linear

τ(z · v) = z · τ(v)

(This is NOT a C2-equivariant complex vector bundle!)

I KR(X ): Grothendieck’s construction

Atiyah’s Real K-theory KR

I C2: cyclic group of order 2 with generator τ

I Real space X : X with a C2-action

Xτ //

id

77Xτ // X

I Real vector bundle E over X :I E : complex vector bundle over XI E : Real spaceI p : E → X is C2-equivariantI τ : Ex → Eτ(x) is anti C-linear

τ(z · v) = z · τ(v)

(This is NOT a C2-equivariant complex vector bundle!)

I KR(X ): Grothendieck’s construction

Atiyah’s Real K-theory KR

I C2: cyclic group of order 2 with generator τ

I Real space X : X with a C2-action

Xτ //

id

77Xτ // X

I Real vector bundle E over X :I E : complex vector bundle over XI E : Real spaceI p : E → X is C2-equivariantI τ : Ex → Eτ(x) is anti C-linear

τ(z · v) = z · τ(v)

(This is NOT a C2-equivariant complex vector bundle!)

I KR(X ): Grothendieck’s construction

Landweber’s Real bordism theory MUR

I γn: the universal bundle over BU(n)

I BU(n)γn : its Thom space

I Σ2BU(n)γn → BU(n + 1)γn+1

=⇒ Thom spectrum MU

I BU(n): Real spaceγn: Real vector bundle

I ΣρBU(n)γn → BU(n + 1)γn+1

ρ: regular representation of C2

=⇒ C2-equivariant Thom spectrum MUR

Landweber’s Real bordism theory MUR

I γn: the universal bundle over BU(n)

I BU(n)γn : its Thom space

I Σ2BU(n)γn → BU(n + 1)γn+1

=⇒ Thom spectrum MU

I BU(n): Real spaceγn: Real vector bundle

I ΣρBU(n)γn → BU(n + 1)γn+1

ρ: regular representation of C2

=⇒ C2-equivariant Thom spectrum MUR

Landweber’s Real bordism theory MUR

I γn: the universal bundle over BU(n)

I BU(n)γn : its Thom space

I Σ2BU(n)γn → BU(n + 1)γn+1

=⇒ Thom spectrum MU

I BU(n): Real spaceγn: Real vector bundle

I ΣρBU(n)γn → BU(n + 1)γn+1

ρ: regular representation of C2

=⇒ C2-equivariant Thom spectrum MUR

Landweber’s Real bordism theory MUR

I γn: the universal bundle over BU(n)

I BU(n)γn : its Thom space

I Σ2BU(n)γn → BU(n + 1)γn+1

=⇒ Thom spectrum MU

I BU(n): Real spaceγn: Real vector bundle

I ΣρBU(n)γn → BU(n + 1)γn+1

ρ: regular representation of C2

=⇒ C2-equivariant Thom spectrum MUR

Landweber’s Real bordism theory MUR

I γn: the universal bundle over BU(n)

I BU(n)γn : its Thom space

I Σ2BU(n)γn → BU(n + 1)γn+1

=⇒ Thom spectrum MU

I BU(n): Real spaceγn: Real vector bundle

I ΣρBU(n)γn → BU(n + 1)γn+1

ρ: regular representation of C2

=⇒ C2-equivariant Thom spectrum MUR

Landweber’s Real bordism theory MUR

I γn: the universal bundle over BU(n)

I BU(n)γn : its Thom space

I Σ2BU(n)γn → BU(n + 1)γn+1

=⇒ Thom spectrum MU

I BU(n): Real spaceγn: Real vector bundle

I ΣρBU(n)γn → BU(n + 1)γn+1

ρ: regular representation of C2

=⇒ C2-equivariant Thom spectrum MUR

Landweber’s Real bordism theory MUR

I γn: the universal bundle over BU(n)

I BU(n)γn : its Thom space

I Σ2BU(n)γn → BU(n + 1)γn+1

=⇒ Thom spectrum MU

I BU(n): Real spaceγn: Real vector bundle

I ΣρBU(n)γn → BU(n + 1)γn+1

ρ: regular representation of C2

=⇒ C2-equivariant Thom spectrum MUR

KR and MUR

I KR and MUR are both C2-equivariant spectra

I Their underlying spectra are KU and MU

I KC2R : KO

MUC2R : harder to describe (computed by Hu–Kriz)

I πu∗MUR carries the universal C2-equivariant formal group law:the C2-action corresponds to the [−1]-series

I Localize at the prime 2,MUR splits as a wedge of suspensions BPR

KR and MUR

I KR and MUR are both C2-equivariant spectra

I Their underlying spectra are KU and MU

I KC2R : KO

MUC2R : harder to describe (computed by Hu–Kriz)

I πu∗MUR carries the universal C2-equivariant formal group law:the C2-action corresponds to the [−1]-series

I Localize at the prime 2,MUR splits as a wedge of suspensions BPR

KR and MUR

I KR and MUR are both C2-equivariant spectra

I Their underlying spectra are KU and MU

I KC2R : KO

MUC2R : harder to describe (computed by Hu–Kriz)

I πu∗MUR carries the universal C2-equivariant formal group law:the C2-action corresponds to the [−1]-series

I Localize at the prime 2,MUR splits as a wedge of suspensions BPR

KR and MUR

I KR and MUR are both C2-equivariant spectra

I Their underlying spectra are KU and MU

I KC2R : KO

MUC2R : harder to describe (computed by Hu–Kriz)

I πu∗MUR carries the universal C2-equivariant formal group law:the C2-action corresponds to the [−1]-series

I Localize at the prime 2,MUR splits as a wedge of suspensions BPR

KR and MUR

I KR and MUR are both C2-equivariant spectra

I Their underlying spectra are KU and MU

I KC2R : KO

MUC2R : harder to describe (computed by Hu–Kriz)

I πu∗MUR carries the universal C2-equivariant formal group law:the C2-action corresponds to the [−1]-series

I Localize at the prime 2,MUR splits as a wedge of suspensions BPR

KR and MUR

I KR and MUR are both C2-equivariant spectra

I Their underlying spectra are KU and MU

I KC2R : KO

MUC2R : harder to describe (computed by Hu–Kriz)

I πu∗MUR carries the universal C2-equivariant formal group law:the C2-action corresponds to the [−1]-series

I Localize at the prime 2,MUR splits as a wedge of suspensions BPR

Extra Features

X : C4-spectrum.

πC4n (X ) = [C4/C4+ ∧ Sn,X ]C4

πC2n (X ) = [C4/C2+ ∧ Sn,X ]C4

πun(X ) = [C4/e+ ∧ Sn,X ]C4

res

res

tr

tr

Mackey functors! πnXtr(res(a)) = 2aMackey functors form an Abelian category!

Extra Features

X : C4-spectrum.

πC4n (X ) = [Sn,X ]C4 = [C4/C4+ ∧ Sn,X ]C4

πC2n (X ) = [C4/C2+ ∧ Sn,X ]C4

πun(X ) = [C4/e+ ∧ Sn,X ]C4

res

res

tr

tr

Mackey functors! πnXtr(res(a)) = 2aMackey functors form an Abelian category!

Extra Features

X : C4-spectrum.

πC4n (X ) = [Sn,X ]C4 = [C4/C4+ ∧ Sn,X ]C4

πC2n (X ) = [Sn,X ]C2 = [C4/C2+ ∧ Sn,X ]C4

πun(X ) = [C4/e+ ∧ Sn,X ]C4

res

res

tr

tr

Mackey functors! πnXtr(res(a)) = 2aMackey functors form an Abelian category!

Extra Features

X : C4-spectrum.

πC4n (X ) = [Sn,X ]C4 = [C4/C4+ ∧ Sn,X ]C4

πC2n (X ) = [Sn,X ]C2 = [C4/C2+ ∧ Sn,X ]C4

πun(X ) = [Sn,X ] = [C4/e+ ∧ Sn,X ]C4

res

res

tr

tr

Mackey functors! πnXtr(res(a)) = 2aMackey functors form an Abelian category!

Extra Features

X : C4-spectrum.

πC4n (X ) = [Sn,X ]C4 = [C4/C4+ ∧ Sn,X ]C4

πC2n (X ) = [Sn,X ]C2 = [C4/C2+ ∧ Sn,X ]C4

πun(X ) = [Sn,X ] = [C4/e+ ∧ Sn,X ]C4

res

res

tr

tr

Mackey functors! πnXtr(res(a)) = 2aMackey functors form an Abelian category!

Extra Features

X : C4-spectrum.

πC4n (X ) = [Sn,X ]C4 = [C4/C4+ ∧ Sn,X ]C4

πC2n (X ) = [Sn,X ]C2 = [C4/C2+ ∧ Sn,X ]C4

πun(X ) = [Sn,X ] = [C4/e+ ∧ Sn,X ]C4

res

res

tr

tr

I Mackey functors! πnX

Extra Features

X : C4-spectrum.

πC4n (X ) = [Sn,X ]C4 = [C4/C4+ ∧ Sn,X ]C4

πC2n (X ) = [Sn,X ]C2 = [C4/C2+ ∧ Sn,X ]C4

πun(X ) = [Sn,X ] = [C4/e+ ∧ Sn,X ]C4

res

res

tr

tr

I Mackey functors! πnX

I tr(res(a)) = 2a

Extra Features

X : C4-spectrum.

πC4n (X ) = [Sn,X ]C4 = [C4/C4+ ∧ Sn,X ]C4

πC2n (X ) = [Sn,X ]C2 = [C4/C2+ ∧ Sn,X ]C4

πun(X ) = [Sn,X ] = [C4/e+ ∧ Sn,X ]C4

res

res

tr

tr

I Mackey functors! πnX

I tr(res(a)) = 2a

I Mackey functors form an Abelian category!

Extra Features

I Non-equivariantly, πnX = [Sn,X ]

I Equivariantly, we have more spheres

I V : G -representation =⇒ SV

I X : G -spectrum

πGVX = [SV ,X ]G

I πGFX : RO(G )-graded homotopy groups of X

I Both at the same time: πF(X )

Extra Features

I Non-equivariantly, πnX = [Sn,X ]

I Equivariantly, we have more spheres

I V : G -representation =⇒ SV

I X : G -spectrum

πGVX = [SV ,X ]G

I πGFX : RO(G )-graded homotopy groups of X

I Both at the same time: πF(X )

Extra Features

I Non-equivariantly, πnX = [Sn,X ]

I Equivariantly, we have more spheres

I V : G -representation =⇒ SV

I X : G -spectrum

πGVX = [SV ,X ]G

I πGFX : RO(G )-graded homotopy groups of X

I Both at the same time: πF(X )

Extra Features

I Non-equivariantly, πnX = [Sn,X ]

I Equivariantly, we have more spheres

I V : G -representation =⇒ SV

I X : G -spectrum

πGVX = [SV ,X ]G

I πGFX : RO(G )-graded homotopy groups of X

I Both at the same time: πF(X )

Extra Features

I Non-equivariantly, πnX = [Sn,X ]

I Equivariantly, we have more spheres

I V : G -representation =⇒ SV

I X : G -spectrum

πGVX = [SV ,X ]G

I πGFX : RO(G )-graded homotopy groups of X

I Both at the same time: πF(X )

Extra Features

I Non-equivariantly, πnX = [Sn,X ]

I Equivariantly, we have more spheres

I V : G -representation =⇒ SV

I X : G -spectrum

πGVX = [SV ,X ]G

I πGFX : RO(G )-graded homotopy groups of X

I Both at the same time: πF(X )

Brown–Peterson and Johnson–Wilson theories

Classically:

BP −→ · · · −→ BP〈2〉 −→ BP〈1〉 −→ BP〈0〉

I π∗BP〈n〉 = Z(2)[v1, . . . , vn]

I BP〈0〉 = HZI BP〈1〉 = ku connective K-theory

I v−1n BP〈n〉 = E (n) Johnson–Wilson theory

Brown–Peterson and Johnson–Wilson theories

Classically:

BP −→ · · · −→ BP〈2〉 −→ BP〈1〉 −→ BP〈0〉

I π∗BP〈n〉 = Z(2)[v1, . . . , vn]

I BP〈0〉 = HZI BP〈1〉 = ku connective K-theory

I v−1n BP〈n〉 = E (n) Johnson–Wilson theory

Brown–Peterson and Johnson–Wilson theories

Classically:

BP −→ · · · −→ BP〈2〉 −→ BP〈1〉 −→ BP〈0〉

I π∗BP〈n〉 = Z(2)[v1, . . . , vn]

I BP〈0〉 = HZ

I BP〈1〉 = ku connective K-theory

I v−1n BP〈n〉 = E (n) Johnson–Wilson theory

Brown–Peterson and Johnson–Wilson theories

Classically:

BP −→ · · · −→ BP〈2〉 −→ BP〈1〉 −→ BP〈0〉

I π∗BP〈n〉 = Z(2)[v1, . . . , vn]

I BP〈0〉 = HZI BP〈1〉 = ku connective K-theory

I v−1n BP〈n〉 = E (n) Johnson–Wilson theory

Brown–Peterson and Johnson–Wilson theories

Classically:

BP −→ · · · −→ BP〈2〉 −→ BP〈1〉 −→ BP〈0〉

I π∗BP〈n〉 = Z(2)[v1, . . . , vn]

I BP〈0〉 = HZI BP〈1〉 = ku connective K-theory

I v−1n BP〈n〉 = E (n) Johnson–Wilson theory

Real Brown–Peterson and Real Johnson–Wilson theories

C2-equivariantly:

BPR −→ · · · −→ BPR〈2〉 −→ BPR〈1〉 −→ BPR〈0〉

I πu∗BPR〈n〉 = Z(2)[v1, . . . , vn]

I BPR〈0〉 = HZ (Hu–Kriz)Z: constant Mackey functor

I BPR〈1〉 = kRI vn ∈ π2(2n−1)BP lifts to vn ∈ πC2

(2n−1)ρBPR

I v−1n BPR〈n〉 = ER(n) Real Johnson–Wilson theory

Real Brown–Peterson and Real Johnson–Wilson theories

C2-equivariantly:

BPR −→ · · · −→ BPR〈2〉 −→ BPR〈1〉 −→ BPR〈0〉

I πu∗BPR〈n〉 = Z(2)[v1, . . . , vn]

I BPR〈0〉 = HZ (Hu–Kriz)Z: constant Mackey functor

I BPR〈1〉 = kRI vn ∈ π2(2n−1)BP lifts to vn ∈ πC2

(2n−1)ρBPR

I v−1n BPR〈n〉 = ER(n) Real Johnson–Wilson theory

Real Brown–Peterson and Real Johnson–Wilson theories

C2-equivariantly:

BPR −→ · · · −→ BPR〈2〉 −→ BPR〈1〉 −→ BPR〈0〉

I πu∗BPR〈n〉 = Z(2)[v1, . . . , vn]

I BPR〈0〉 = HZ (Hu–Kriz)Z: constant Mackey functor

I BPR〈1〉 = kRI vn ∈ π2(2n−1)BP lifts to vn ∈ πC2

(2n−1)ρBPR

I v−1n BPR〈n〉 = ER(n) Real Johnson–Wilson theory

Real Brown–Peterson and Real Johnson–Wilson theories

C2-equivariantly:

BPR −→ · · · −→ BPR〈2〉 −→ BPR〈1〉 −→ BPR〈0〉

I πu∗BPR〈n〉 = Z(2)[v1, . . . , vn]

I BPR〈0〉 = HZ (Hu–Kriz)Z: constant Mackey functor

I BPR〈1〉 = kR

I vn ∈ π2(2n−1)BP lifts to vn ∈ πC2

(2n−1)ρBPR

I v−1n BPR〈n〉 = ER(n) Real Johnson–Wilson theory

Real Brown–Peterson and Real Johnson–Wilson theories

C2-equivariantly:

BPR −→ · · · −→ BPR〈2〉 −→ BPR〈1〉 −→ BPR〈0〉

I πu∗BPR〈n〉 = Z(2)[v1, . . . , vn]

I BPR〈0〉 = HZ (Hu–Kriz)Z: constant Mackey functor

I BPR〈1〉 = kRI vn ∈ π2(2n−1)BP lifts to vn ∈ πC2

(2n−1)ρBPR

I v−1n BPR〈n〉 = ER(n) Real Johnson–Wilson theory

Real Brown–Peterson and Real Johnson–Wilson theories

C2-equivariantly:

BPR −→ · · · −→ BPR〈2〉 −→ BPR〈1〉 −→ BPR〈0〉

I πu∗BPR〈n〉 = Z(2)[v1, . . . , vn]

I BPR〈0〉 = HZ (Hu–Kriz)Z: constant Mackey functor

I BPR〈1〉 = kRI vn ∈ π2(2n−1)BP lifts to vn ∈ πC2

(2n−1)ρBPR

I v−1n BPR〈n〉 = ER(n) Real Johnson–Wilson theory

The Hill–Hopkins–Ravenel Norm

I NC2m

C2: C2-Spectra −→ C2m -Spectra

I MU((C2m )) := NC2m

C2(MUR)

underlying spectrum of MU((C8)): MU ∧MU ∧MU ∧MU(a, b, c , d) −→ (d , a, b, c)

I πu∗MU((C2m )) carries the universal C2m -equivariant formalgroup law such that the C2-action corresponds to the[−1]-series

I BP((C2m )) := NC2m

C2(BPR)

MU((C2m )) splits as a wedge of suspensions of BP((C2m ))

The Hill–Hopkins–Ravenel Norm

I NC2m

C2: C2-Spectra −→ C2m -Spectra

I MU((C2m )) := NC2m

C2(MUR)

underlying spectrum of MU((C8)): MU ∧MU ∧MU ∧MU(a, b, c , d) −→ (d , a, b, c)

I πu∗MU((C2m )) carries the universal C2m -equivariant formalgroup law such that the C2-action corresponds to the[−1]-series

I BP((C2m )) := NC2m

C2(BPR)

MU((C2m )) splits as a wedge of suspensions of BP((C2m ))

The Hill–Hopkins–Ravenel Norm

I NC2m

C2: C2-Spectra −→ C2m -Spectra

I MU((C2m )) := NC2m

C2(MUR)

underlying spectrum of MU((C8)): MU ∧MU ∧MU ∧MU

(a, b, c , d) −→ (d , a, b, c)

I πu∗MU((C2m )) carries the universal C2m -equivariant formalgroup law such that the C2-action corresponds to the[−1]-series

I BP((C2m )) := NC2m

C2(BPR)

MU((C2m )) splits as a wedge of suspensions of BP((C2m ))

The Hill–Hopkins–Ravenel Norm

I NC2m

C2: C2-Spectra −→ C2m -Spectra

I MU((C2m )) := NC2m

C2(MUR)

underlying spectrum of MU((C8)): MU ∧MU ∧MU ∧MU(a, b, c , d) −→ (d , a, b, c)

I πu∗MU((C2m )) carries the universal C2m -equivariant formalgroup law such that the C2-action corresponds to the[−1]-series

I BP((C2m )) := NC2m

C2(BPR)

MU((C2m )) splits as a wedge of suspensions of BP((C2m ))

The Hill–Hopkins–Ravenel Norm

I NC2m

C2: C2-Spectra −→ C2m -Spectra

I MU((C2m )) := NC2m

C2(MUR)

underlying spectrum of MU((C8)): MU ∧MU ∧MU ∧MU(a, b, c , d) −→ (d , a, b, c)

I πu∗MU((C2m )) carries the universal C2m -equivariant formalgroup law such that the C2-action corresponds to the[−1]-series

I BP((C2m )) := NC2m

C2(BPR)

MU((C2m )) splits as a wedge of suspensions of BP((C2m ))

The Hill–Hopkins–Ravenel Norm

I NC2m

C2: C2-Spectra −→ C2m -Spectra

I MU((C2m )) := NC2m

C2(MUR)

underlying spectrum of MU((C8)): MU ∧MU ∧MU ∧MU(a, b, c , d) −→ (d , a, b, c)

I πu∗MU((C2m )) carries the universal C2m -equivariant formalgroup law such that the C2-action corresponds to the[−1]-series

I BP((C2m )) := NC2m

C2(BPR)

MU((C2m )) splits as a wedge of suspensions of BP((C2m ))

Hill–Hopkins–Ravenel Theorems

Theorem (Hill–Hopkins–Ravenel)

For j ≥ 7, the Kervaire invariant elements θj do not exist.

I Start with MU((C8))

I Invert a certain class D ∈ πC8F MU((C8)): D−1MU((C8))

I Ω: its C8-fixed point spectrum

I Detection Theorem:If θj exists, then its image in π2j+1−2Ω is nonzero.

I Periodicity Theorem:π∗Ω is 256-periodic.

I Gap Theorem:πiΩ = 0 for i = −1,−2,−3.

Hill–Hopkins–Ravenel Theorems

Theorem (Hill–Hopkins–Ravenel)

For j ≥ 7, the Kervaire invariant elements θj do not exist.

I Start with MU((C8))

I Invert a certain class D ∈ πC8F MU((C8)): D−1MU((C8))

I Ω: its C8-fixed point spectrum

I Detection Theorem:If θj exists, then its image in π2j+1−2Ω is nonzero.

I Periodicity Theorem:π∗Ω is 256-periodic.

I Gap Theorem:πiΩ = 0 for i = −1,−2,−3.

Hill–Hopkins–Ravenel Theorems

Theorem (Hill–Hopkins–Ravenel)

For j ≥ 7, the Kervaire invariant elements θj do not exist.

I Start with MU((C8))

I Invert a certain class D ∈ πC8F MU((C8)): D−1MU((C8))

I Ω: its C8-fixed point spectrum

I Detection Theorem:If θj exists, then its image in π2j+1−2Ω is nonzero.

I Periodicity Theorem:π∗Ω is 256-periodic.

I Gap Theorem:πiΩ = 0 for i = −1,−2,−3.

Hill–Hopkins–Ravenel Theorems

Theorem (Hill–Hopkins–Ravenel)

For j ≥ 7, the Kervaire invariant elements θj do not exist.

I Start with MU((C8))

I Invert a certain class D ∈ πC8F MU((C8)): D−1MU((C8))

I Ω: its C8-fixed point spectrum

I Detection Theorem:If θj exists, then its image in π2j+1−2Ω is nonzero.

I Periodicity Theorem:π∗Ω is 256-periodic.

I Gap Theorem:πiΩ = 0 for i = −1,−2,−3.

Hill–Hopkins–Ravenel Theorems

Theorem (Hill–Hopkins–Ravenel)

For j ≥ 7, the Kervaire invariant elements θj do not exist.

I Start with MU((C8))

I Invert a certain class D ∈ πC8F MU((C8)): D−1MU((C8))

I Ω: its C8-fixed point spectrum

I Detection Theorem:If θj exists, then its image in π2j+1−2Ω is nonzero.

I Periodicity Theorem:π∗Ω is 256-periodic.

I Gap Theorem:πiΩ = 0 for i = −1,−2,−3.

Hill–Hopkins–Ravenel Theorems

Theorem (Hill–Hopkins–Ravenel)

For j ≥ 7, the Kervaire invariant elements θj do not exist.

I Start with MU((C8))

I Invert a certain class D ∈ πC8F MU((C8)): D−1MU((C8))

I Ω: its C8-fixed point spectrum

I Detection Theorem:If θj exists, then its image in π2j+1−2Ω is nonzero.

I Periodicity Theorem:π∗Ω is 256-periodic.

I Gap Theorem:πiΩ = 0 for i = −1,−2,−3.

Hill–Hopkins–Ravenel Theorems

Theorem (Hill–Hopkins–Ravenel)

For j ≥ 7, the Kervaire invariant elements θj do not exist.

I Start with MU((C8))

I Invert a certain class D ∈ πC8F MU((C8)): D−1MU((C8))

I Ω: its C8-fixed point spectrum

I Detection Theorem:If θj exists, then its image in π2j+1−2Ω is nonzero.

I Periodicity Theorem:π∗Ω is 256-periodic.

I Gap Theorem:πiΩ = 0 for i = −1,−2,−3.

Baby Ω

Gap Theorem

Periodicity Theorem

Hill–Hopkins–Ravenel Theories

BP((C2m )) → · · · → BP((C2m ))〈2〉 → BP((C2m ))〈1〉 → BP((C2m ))〈0〉

I πu∗BP((C2m ))〈n〉 = Z(2)[C2m · r1,C2m · r3, . . . ,C2m · r2n−1]

γ: generator of C2m

C2m · ri := ri , γri , . . . , γ2m−1−1ri

I BP((C2m ))〈0〉 = HZ(Hill–Hopkins–Ravenel reduction theorem)

I (BP((C8))〈1〉)C8 also detects θj !

Hill–Hopkins–Ravenel Theories

BP((C2m )) → · · · → BP((C2m ))〈2〉 → BP((C2m ))〈1〉 → BP((C2m ))〈0〉

I πu∗BP((C2m ))〈n〉 = Z(2)[C2m · r1,C2m · r3, . . . ,C2m · r2n−1]

γ: generator of C2m

C2m · ri := ri , γri , . . . , γ2m−1−1ri

I BP((C2m ))〈0〉 = HZ(Hill–Hopkins–Ravenel reduction theorem)

I (BP((C8))〈1〉)C8 also detects θj !

Hill–Hopkins–Ravenel Theories

BP((C2m )) → · · · → BP((C2m ))〈2〉 → BP((C2m ))〈1〉 → BP((C2m ))〈0〉

I πu∗BP((C2m ))〈n〉 = Z(2)[C2m · r1,C2m · r3, . . . ,C2m · r2n−1]

γ: generator of C2m

C2m · ri := ri , γri , . . . , γ2m−1−1ri

I BP((C2m ))〈0〉 = HZ(Hill–Hopkins–Ravenel reduction theorem)

I (BP((C8))〈1〉)C8 also detects θj !

Hill–Hopkins–Ravenel Theories

BP((C2m )) → · · · → BP((C2m ))〈2〉 → BP((C2m ))〈1〉 → BP((C2m ))〈0〉

I πu∗BP((C2m ))〈n〉 = Z(2)[C2m · r1,C2m · r3, . . . ,C2m · r2n−1]

γ: generator of C2m

C2m · ri := ri , γri , . . . , γ2m−1−1ri

I BP((C2m ))〈0〉 = HZ(Hill–Hopkins–Ravenel reduction theorem)

I (BP((C8))〈1〉)C8 also detects θj !

Hill–Hopkins–Ravenel Theories

BP((C2m )) → · · · → BP((C2m ))〈2〉 → BP((C2m ))〈1〉 → BP((C2m ))〈0〉

I πu∗BP((C2m ))〈n〉 = Z(2)[C2m · r1,C2m · r3, . . . ,C2m · r2n−1]

γ: generator of C2m

C2m · ri := ri , γri , . . . , γ2m−1−1ri

I BP((C2m ))〈0〉 = HZ(Hill–Hopkins–Ravenel reduction theorem)

I (BP((C8))〈1〉)C8 also detects θj !

Slice Cells and the Slice Tower

I G : finite group. H ⊂ G . ρH : regular representation of H

I Slice cells:G+ ∧H SnρH and G+ ∧H SnρH−1

Dimension = underlying dimensionI S>n: smallest full subcategory of G -spectra that

I contains slice cells of dimension > nI closed under taking cofibers, extensions and wedges

I S≤n = X | MapG (Y ,X ) ' ∗,Y ∈ S>nI Pn+1X −→ X −→ PnX

Pn+1X ∈ S>n, PnX ∈ S≤n

Slice Cells and the Slice Tower

I G : finite group. H ⊂ G . ρH : regular representation of H

I Slice cells:G+ ∧H SnρH and G+ ∧H SnρH−1

Dimension = underlying dimensionI S>n: smallest full subcategory of G -spectra that

I contains slice cells of dimension > nI closed under taking cofibers, extensions and wedges

I S≤n = X | MapG (Y ,X ) ' ∗,Y ∈ S>nI Pn+1X −→ X −→ PnX

Pn+1X ∈ S>n, PnX ∈ S≤n

Slice Cells and the Slice Tower

I G : finite group. H ⊂ G . ρH : regular representation of H

I Slice cells:G+ ∧H SnρH and G+ ∧H SnρH−1

Dimension = underlying dimension

I S>n: smallest full subcategory of G -spectra thatI contains slice cells of dimension > nI closed under taking cofibers, extensions and wedges

I S≤n = X | MapG (Y ,X ) ' ∗,Y ∈ S>nI Pn+1X −→ X −→ PnX

Pn+1X ∈ S>n, PnX ∈ S≤n

Slice Cells and the Slice Tower

I G : finite group. H ⊂ G . ρH : regular representation of H

I Slice cells:G+ ∧H SnρH and G+ ∧H SnρH−1

Dimension = underlying dimensionI S>n: smallest full subcategory of G -spectra that

I contains slice cells of dimension > n

I closed under taking cofibers, extensions and wedges

I S≤n = X | MapG (Y ,X ) ' ∗,Y ∈ S>nI Pn+1X −→ X −→ PnX

Pn+1X ∈ S>n, PnX ∈ S≤n

Slice Cells and the Slice Tower

I G : finite group. H ⊂ G . ρH : regular representation of H

I Slice cells:G+ ∧H SnρH and G+ ∧H SnρH−1

Dimension = underlying dimensionI S>n: smallest full subcategory of G -spectra that

I contains slice cells of dimension > nI closed under taking cofibers, extensions and wedges

I S≤n = X | MapG (Y ,X ) ' ∗,Y ∈ S>nI Pn+1X −→ X −→ PnX

Pn+1X ∈ S>n, PnX ∈ S≤n

Slice Cells and the Slice Tower

I G : finite group. H ⊂ G . ρH : regular representation of H

I Slice cells:G+ ∧H SnρH and G+ ∧H SnρH−1

Dimension = underlying dimensionI S>n: smallest full subcategory of G -spectra that

I contains slice cells of dimension > nI closed under taking cofibers, extensions and wedges

I S≤n = X | MapG (Y ,X ) ' ∗,Y ∈ S>n

I Pn+1X −→ X −→ PnXPn+1X ∈ S>n, P

nX ∈ S≤n

Slice Cells and the Slice Tower

I G : finite group. H ⊂ G . ρH : regular representation of H

I Slice cells:G+ ∧H SnρH and G+ ∧H SnρH−1

Dimension = underlying dimensionI S>n: smallest full subcategory of G -spectra that

I contains slice cells of dimension > nI closed under taking cofibers, extensions and wedges

I S≤n = X | MapG (Y ,X ) ' ∗,Y ∈ S>nI Pn+1X −→ X −→ PnX

Pn+1X ∈ S>n, PnX ∈ S≤n

Slice tower:

X · · · PnX Pn−1X · · ·

PnnX Pn−1

n−1X

Apply πG∗ (−):

πG∗ (X ) · · · πG∗ (PnX ) πG∗ (Pn−1X ) · · ·

πG∗ (PnnX ) πG∗ (Pn−1

n−1X )

E s,t1 = πGt−sP

ttX =⇒ πGt−sX

dr : E s,tr −→ E s+r ,t+r−1

r

Slice tower:

X · · · PnX Pn−1X · · ·

PnnX Pn−1

n−1X

Apply πG∗ (−):

πG∗ (X ) · · · πG∗ (PnX ) πG∗ (Pn−1X ) · · ·

πG∗ (PnnX ) πG∗ (Pn−1

n−1X )

E s,t1 = πGt−sP

ttX =⇒ πGt−sX

dr : E s,tr −→ E s+r ,t+r−1

r

Slice tower:

X · · · PnX Pn−1X · · ·

PnnX Pn−1

n−1X

Apply πG∗ (−):

πG∗ (X ) · · · πG∗ (PnX ) πG∗ (Pn−1X ) · · ·

πG∗ (PnnX ) πG∗ (Pn−1

n−1X )

E s,t1 = πGt−sP

ttX =⇒ πGt−sX

dr : E s,tr −→ E s+r ,t+r−1

r

Slice tower:

X · · · PnX Pn−1X · · ·

PnnX Pn−1

n−1X

Apply πGF(−):

πGF(X ) · · · πGF(PnX ) πGF(Pn−1X ) · · ·

πGF(PnnX ) πGF(Pn−1

n−1X )

E s,t1 = πGV−sP

|V ||V |X =⇒ πGV−sX

dr : E s,Vr −→ E s+r ,V+r−1

r

Slice tower:

X · · · PnX Pn−1X · · ·

PnnX Pn−1

n−1X

Apply πGF(−):

πGF(X ) · · · πGF(PnX ) πGF(Pn−1X ) · · ·

πGF(PnnX ) πGF(Pn−1

n−1X )

E s,t1 = πGV−sP

|V ||V |X =⇒ πGV−sX

dr : E s,Vr −→ E s+r ,V+r−1

r

Slice tower:

X · · · PnX Pn−1X · · ·

PnnX Pn−1

n−1X

Apply πGF(−):

πGF(X ) · · · πGF(PnX ) πGF(Pn−1X ) · · ·

πGF(PnnX ) πGF(Pn−1

n−1X )

E s,t1 = πGV−sP

|V ||V |X =⇒ πGV−sX

dr : E s,Vr −→ E s+r ,V+r−1

r

Slice tower:

X · · · PnX Pn−1X · · ·

PnnX Pn−1

n−1X

Apply π∗(−):

π∗(X ) · · · π∗(PnX ) π∗(P

n−1X ) · · ·

π∗(PnnX ) π∗(P

n−1n−1X )

E s,t1 = πt−sP

ttX =⇒ πt−sX

dr : E s,tr −→ E s+r ,t+r−1

r

Slice tower:

X · · · PnX Pn−1X · · ·

PnnX Pn−1

n−1X

Apply π∗(−):

π∗(X ) · · · π∗(PnX ) π∗(P

n−1X ) · · ·

π∗(PnnX ) π∗(P

n−1n−1X )

E s,t1 = πt−sP

ttX =⇒ πt−sX

dr : E s,tr −→ E s+r ,t+r−1

r

Slice tower:

X · · · PnX Pn−1X · · ·

PnnX Pn−1

n−1X

Apply π∗(−):

π∗(X ) · · · π∗(PnX ) π∗(P

n−1X ) · · ·

π∗(PnnX ) π∗(P

n−1n−1X )

E s,t1 = πt−sP

ttX =⇒ πt−sX

dr : E s,tr −→ E s+r ,t+r−1

r

This is a spectral sequence of Mackey functors.

It is also a Mackey functor of spectral sequences!

This is a spectral sequence of Mackey functors.

It is also a Mackey functor of spectral sequences!

Slice tower:

X · · · PnX Pn−1X · · ·

PnnX Pn−1

n−1X

πC4∗ (X ) · · · πC4

∗ (PnX ) πC4∗ (Pn−1X ) · · ·

πC4∗ (Pn

nX ) πC4∗ (Pn−1

n−1X )

πC2∗ (X ) · · · πC2

∗ (PnX ) πC2∗ (Pn−1X ) · · ·

πC2∗ (Pn

nX ) πC2∗ (Pn−1

n−1X )

πC4∗ (X ) · · · πC4

∗ (PnX ) πC4∗ (Pn−1X ) · · ·

πC4∗ (Pn

nX ) πC4∗ (Pn−1

n−1X )

πC2∗ (X ) · · · πC2

∗ (PnX ) πC2∗ (Pn−1X ) · · ·

πC2∗ (Pn

nX ) πC2∗ (Pn−1

n−1X )

πC4∗ (X ) · · · πC4

∗ (PnX ) πC4∗ (Pn−1X ) · · ·

πC4∗ (Pn

nX ) πC4∗ (Pn−1

n−1X )

πC2∗ (X ) · · · πC2

∗ (PnX ) πC2∗ (Pn−1X ) · · ·

πC2∗ (Pn

nX ) πC2∗ (Pn−1

n−1X )

restr

πC4∗ (X ) · · · πC4

∗ (PnX ) πC4∗ (Pn−1X ) · · ·

πC4∗ (Pn

nX ) πC4∗ (Pn−1

n−1X )

πC2∗ (X ) · · · πC2

∗ (PnX ) πC2∗ (Pn−1X ) · · ·

πC2∗ (Pn

nX ) πC2∗ (Pn−1

n−1X )

restr

Slice tower:

X · · · PnX Pn−1X · · ·

PnnX Pn−1

n−1X

Apply πF(−):

πF(X ) · · · πF(PnX ) πF(Pn−1X ) · · ·

πF(PnnX ) πF(Pn−1

n−1X )

E s,V1 = πV−sP

|V ||V |X =⇒ πV−sX

dr : E s,Vr −→ E s+r ,V+r−1

r

Slice tower:

X · · · PnX Pn−1X · · ·

PnnX Pn−1

n−1X

Apply πF(−):

πF(X ) · · · πF(PnX ) πF(Pn−1X ) · · ·

πF(PnnX ) πF(Pn−1

n−1X )

E s,V1 = πV−sP

|V ||V |X =⇒ πV−sX

dr : E s,Vr −→ E s+r ,V+r−1

r

Slice tower:

X · · · PnX Pn−1X · · ·

PnnX Pn−1

n−1X

Apply πF(−):

πF(X ) · · · πF(PnX ) πF(Pn−1X ) · · ·

πF(PnnX ) πF(Pn−1

n−1X )

E s,V1 = πV−sP

|V ||V |X =⇒ πV−sX

dr : E s,Vr −→ E s+r ,V+r−1

r

C2-SliceSS(KR): π∗KR

−16 −12 −8 −4 0 4 8 12 16

−8

−4

0

4

8

C2-SliceSS(KR)

I Dugger computed the C2-slice spectral sequence of KR

I P2n−12n−1KR = 0 for all n ∈ Z

I P2n2nKR = Snρ ∧ HZ for all n ∈ Z

I E1-page:

πC2F (Snρ ∧ HZ)

= HC2F (Snρ;Z)

= πC2F−nρHZ

I Note that RO(C2) = Z⊕ Zgenerated by 1 and σ

I aσ : S0 −→ Sσ

I u2σ: generator of HC22 (S2σ;Z) = Z

C2-SliceSS(KR)

I Dugger computed the C2-slice spectral sequence of KR

I P2n−12n−1KR = 0 for all n ∈ Z

I P2n2nKR = Snρ ∧ HZ for all n ∈ Z

I E1-page:

πC2F (Snρ ∧ HZ)

= HC2F (Snρ;Z)

= πC2F−nρHZ

I Note that RO(C2) = Z⊕ Zgenerated by 1 and σ

I aσ : S0 −→ Sσ

I u2σ: generator of HC22 (S2σ;Z) = Z

C2-SliceSS(KR)

I Dugger computed the C2-slice spectral sequence of KR

I P2n−12n−1KR = 0 for all n ∈ Z

I P2n2nKR = Snρ ∧ HZ for all n ∈ Z

I E1-page:

πC2F (Snρ ∧ HZ)

= HC2F (Snρ;Z)

= πC2F−nρHZ

I Note that RO(C2) = Z⊕ Zgenerated by 1 and σ

I aσ : S0 −→ Sσ

I u2σ: generator of HC22 (S2σ;Z) = Z

C2-SliceSS(KR)

I Dugger computed the C2-slice spectral sequence of KR

I P2n−12n−1KR = 0 for all n ∈ Z

I P2n2nKR = Snρ ∧ HZ for all n ∈ Z

I E1-page:

πC2F (Snρ ∧ HZ)

= HC2F (Snρ;Z)

= πC2F−nρHZ

I Note that RO(C2) = Z⊕ Zgenerated by 1 and σ

I aσ : S0 −→ Sσ

I u2σ: generator of HC22 (S2σ;Z) = Z

C2-SliceSS(KR)

I Dugger computed the C2-slice spectral sequence of KR

I P2n−12n−1KR = 0 for all n ∈ Z

I P2n2nKR = Snρ ∧ HZ for all n ∈ Z

I E1-page:

πC2F (Snρ ∧ HZ)

= HC2F (Snρ;Z)

= πC2F−nρHZ

I Note that RO(C2) = Z⊕ Zgenerated by 1 and σ

I aσ : S0 −→ Sσ

I u2σ: generator of HC22 (S2σ;Z) = Z

C2-SliceSS(KR)

I Dugger computed the C2-slice spectral sequence of KR

I P2n−12n−1KR = 0 for all n ∈ Z

I P2n2nKR = Snρ ∧ HZ for all n ∈ Z

I E1-page:

πC2F (Snρ ∧ HZ)

= HC2F (Snρ;Z)

= πC2F−nρHZ

I Note that RO(C2) = Z⊕ Zgenerated by 1 and σ

I aσ : S0 −→ Sσ

I u2σ: generator of HC22 (S2σ;Z) = Z

C2-SliceSS(KR)

I Dugger computed the C2-slice spectral sequence of KR

I P2n−12n−1KR = 0 for all n ∈ Z

I P2n2nKR = Snρ ∧ HZ for all n ∈ Z

I E1-page:

πC2F (Snρ ∧ HZ)

= HC2F (Snρ;Z)

= πC2F−nρHZ

I Note that RO(C2) = Z⊕ Zgenerated by 1 and σ

I aσ : S0 −→ Sσ

I u2σ: generator of HC22 (S2σ;Z) = Z

C2-SliceSS(KR)

I Dugger computed the C2-slice spectral sequence of KR

I P2n−12n−1KR = 0 for all n ∈ Z

I P2n2nKR = Snρ ∧ HZ for all n ∈ Z

I E1-page:

πC2F (Snρ ∧ HZ)

= HC2F (Snρ;Z)

= πC2F−nρHZ

I Note that RO(C2) = Z⊕ Zgenerated by 1 and σ

I aσ : S0 −→ Sσ

I u2σ: generator of HC22 (S2σ;Z) = Z

C2-SliceSS(KR)

I Dugger computed the C2-slice spectral sequence of KR

I P2n−12n−1KR = 0 for all n ∈ Z

I P2n2nKR = Snρ ∧ HZ for all n ∈ Z

I E1-page:

πC2F (Snρ ∧ HZ)

= HC2F (Snρ;Z)

= πC2F−nρHZ

I Note that RO(C2) = Z⊕ Zgenerated by 1 and σ

I aσ : S0 −→ Sσ

I u2σ: generator of HC22 (S2σ;Z) = Z

πC2

a+bσHZ

−8 −6 −4 −2 0 2 4 6 8

−8

−6

−4

−2

0

2

4

6

8

a

b

2u22σ

2u2σ

1

u2σ

u22σ

aσ

a2σ

a3σ u2σaσ

θ

θaσ

θa2σ

θu2σ

1

C2-SliceSS(KR): π∗KR

−16 −12 −8 −4 0 4 8 12 16

−8

−4

0

4

8

C2-SliceSS(KR): πu∗KR = π∗KU

−16 −12 −8 −4 0 4 8 12 16

−8

−4

0

4

8

C2-SliceSS(KR): πC2∗ KR = π∗KO

−16 −12 −8 −4 0 4 8 12 16

−8

−4

0

4

8

C2-SliceSS(KR): πC2∗ KR = π∗KO

−16 −12 −8 −4 0 4 8 12 16

−8

−4

0

4

8

v1aσ

v21a2σ

v31a3σ

v21u2σ v4

1u22σ

2v21u2σ

θv31

θv41aσ

θv51a2σ

θv61a3σ

2v41u

22σ

θv51u2σ

Two periodicities

In the RO(C2)-grading:

I v1 ∈ πC2ρ KR gives ρ-periodicity

I u22σ ∈ πC24−4σKR gives (4− 4σ)-periodicity

πC2F+ρKR = πC2

F KR

πC2F+8KR = πC2

F KR

Two equivalences:

Sρ ∧ KR ' KR

S8 ∧ KR ' KR

Two periodicities

In the RO(C2)-grading:

I v1 ∈ πC2ρ KR gives ρ-periodicity

I u22σ ∈ πC24−4σKR gives (4− 4σ)-periodicity

πC2F+ρKR = πC2

F KR

πC2F+8KR = πC2

F KR

Two equivalences:

Sρ ∧ KR ' KR

S8 ∧ KR ' KR

Two periodicities

In the RO(C2)-grading:

I v1 ∈ πC2ρ KR gives ρ-periodicity

I u22σ ∈ πC24−4σKR gives (4− 4σ)-periodicity

πC2F+ρKR = πC2

F KR

πC2F+8KR = πC2

F KR

Two equivalences:

Sρ ∧ KR ' KR

S8 ∧ KR ' KR

Two periodicities

In the RO(C2)-grading:

I v1 ∈ πC2ρ KR gives ρ-periodicity

I u22σ ∈ πC24−4σKR gives (4− 4σ)-periodicity

πC2F+ρKR = πC2

F KR

πC2F+8KR = πC2

F KR

Two equivalences:

Sρ ∧ KR ' KR

S8 ∧ KR ' KR

Two periodicities

In the RO(C2)-grading:

I v1 ∈ πC2ρ KR gives ρ-periodicity

I u22σ ∈ πC24−4σKR gives (4− 4σ)-periodicity

πC2F+ρKR = πC2

F KR

πC2F+8KR = πC2

F KR

Two equivalences:

Sρ ∧ KR ' KR

S8 ∧ KR ' KR

Slices of BP ((C2m))〈n〉

I Odd slices ' ∗

I Even slices =∨

H⊆G nontrivial

(G+ ∧H SmρH ) ∧ HZ

I E1-page: HGF(G+ ∧H SmρH ;Z)

Computable!

I What about Detection Theorems, Periodicity Theorems, andGap Theorems?

Slices of BP ((C2m))〈n〉

I Odd slices ' ∗I Even slices =

∨H⊆G nontrivial

(G+ ∧H SmρH ) ∧ HZ

I E1-page: HGF(G+ ∧H SmρH ;Z)

Computable!

I What about Detection Theorems, Periodicity Theorems, andGap Theorems?

Slices of BP ((C2m))〈n〉

I Odd slices ' ∗I Even slices =

∨H⊆G nontrivial

(G+ ∧H SmρH ) ∧ HZ

I E1-page: HGF(G+ ∧H SmρH ;Z)

Computable!

I What about Detection Theorems, Periodicity Theorems, andGap Theorems?

Slices of BP ((C2m))〈n〉

I Odd slices ' ∗I Even slices =

∨H⊆G nontrivial

(G+ ∧H SmρH ) ∧ HZ

I E1-page: HGF(G+ ∧H SmρH ;Z)

Computable!

I What about Detection Theorems, Periodicity Theorems, andGap Theorems?

Slices of BP ((C2m))〈n〉

I Odd slices ' ∗I Even slices =

∨H⊆G nontrivial

(G+ ∧H SmρH ) ∧ HZ

I E1-page: HGF(G+ ∧H SmρH ;Z)

Computable!

I What about Detection Theorems, Periodicity Theorems, andGap Theorems?

Hurewicz Image of BPC2

R

BPR · · · BPR〈3〉 BPR〈2〉 BPR〈1〉

Hurewicz Image of BPC2

R

BPC2R · · · BPR〈3〉C2 BPR〈2〉C2 BPR〈1〉C2

Hurewicz Image of BPC2

R

BPC2R · · · BPR〈3〉C2 BPR〈2〉C2 BPR〈1〉C2

S

What are the Hurewicz images?

Theorem (Li–S.–Wang–Xu)

The Hopf, Kervaire, and κ-families are detected by the Hurewiczmap π∗S→ π∗BP

C2R .

Theorem (Li–S.–Wang–Xu)

The C2-fixed points of BPR〈n〉 detects the first n elements of theHopf- and Kervaire-family, and the first (n − 1) elements of theκ-family.

Theorem (Li–S.–Wang–Xu)

The Hopf, Kervaire, and κ-families are detected by the Hurewiczmap π∗S→ π∗BP

C2R .

Theorem (Li–S.–Wang–Xu)

The C2-fixed points of BPR〈n〉 detects the first n elements of theHopf- and Kervaire-family, and the first (n − 1) elements of theκ-family.

Some classes on the Adams E2-page

I hi ∈ Ext1,2i

A∗(F2,F2)

:hi survives and detects Hopf map for i ≤ 3hi supports nonzero differentials for i ≥ 4

I h2j ∈ Ext2,2j+1

A∗(F2,F2):

h2j survives and detects Kervaire class θj for j ≤ 5

h2j supports differential for j ≥ 7

the fate of h26 is unknown

I gk = 〈h2k+2, hk−1, hk , hk+1〉 ∈ Ext4,2k+2+2k+3

A∗(F2,F2):

g1, g2 survive and detect κ, κ2 in stems 20, 44g3 supports a nonzero differential in stem 92the fate of gk for k ≥ 4 is unknown

Some classes on the Adams E2-page

I hi ∈ Ext1,2i

A∗(F2,F2):

hi survives and detects Hopf map for i ≤ 3hi supports nonzero differentials for i ≥ 4

I h2j ∈ Ext2,2j+1

A∗(F2,F2):

h2j survives and detects Kervaire class θj for j ≤ 5

h2j supports differential for j ≥ 7

the fate of h26 is unknown

I gk = 〈h2k+2, hk−1, hk , hk+1〉 ∈ Ext4,2k+2+2k+3

A∗(F2,F2):

g1, g2 survive and detect κ, κ2 in stems 20, 44g3 supports a nonzero differential in stem 92the fate of gk for k ≥ 4 is unknown

Some classes on the Adams E2-page

I hi ∈ Ext1,2i

A∗(F2,F2):

hi survives and detects Hopf map for i ≤ 3hi supports nonzero differentials for i ≥ 4

I h2j ∈ Ext2,2j+1

A∗(F2,F2):

h2j survives and detects Kervaire class θj for j ≤ 5

h2j supports differential for j ≥ 7

the fate of h26 is unknown

I gk = 〈h2k+2, hk−1, hk , hk+1〉 ∈ Ext4,2k+2+2k+3

A∗(F2,F2):

g1, g2 survive and detect κ, κ2 in stems 20, 44g3 supports a nonzero differential in stem 92the fate of gk for k ≥ 4 is unknown

Some classes on the Adams E2-page

I hi ∈ Ext1,2i

A∗(F2,F2):

hi survives and detects Hopf map for i ≤ 3hi supports nonzero differentials for i ≥ 4

I h2j ∈ Ext2,2j+1

A∗(F2,F2):

h2j survives and detects Kervaire class θj for j ≤ 5

h2j supports differential for j ≥ 7

the fate of h26 is unknown

I gk = 〈h2k+2, hk−1, hk , hk+1〉 ∈ Ext4,2k+2+2k+3

A∗(F2,F2):

g1, g2 survive and detect κ, κ2 in stems 20, 44g3 supports a nonzero differential in stem 92the fate of gk for k ≥ 4 is unknown

Some classes on the Adams E2-page

I hi ∈ Ext1,2i

A∗(F2,F2):

hi survives and detects Hopf map for i ≤ 3hi supports nonzero differentials for i ≥ 4

I h2j ∈ Ext2,2j+1

A∗(F2,F2):

h2j survives and detects Kervaire class θj for j ≤ 5

h2j supports differential for j ≥ 7

the fate of h26 is unknown

I gk = 〈h2k+2, hk−1, hk , hk+1〉 ∈ Ext4,2k+2+2k+3

A∗(F2,F2):

g1, g2 survive and detect κ, κ2 in stems 20, 44g3 supports a nonzero differential in stem 92the fate of gk for k ≥ 4 is unknown

Some classes on the Adams E2-page

I hi ∈ Ext1,2i

A∗(F2,F2):

hi survives and detects Hopf map for i ≤ 3hi supports nonzero differentials for i ≥ 4

I h2j ∈ Ext2,2j+1

A∗(F2,F2):

h2j survives and detects Kervaire class θj for j ≤ 5

h2j supports differential for j ≥ 7

the fate of h26 is unknown

I gk = 〈h2k+2, hk−1, hk , hk+1〉 ∈ Ext4,2k+2+2k+3

A∗(F2,F2):

g1, g2 survive and detect κ, κ2 in stems 20, 44g3 supports a nonzero differential in stem 92the fate of gk for k ≥ 4 is unknown

Some classes on the Adams E2-page

I hi ∈ Ext1,2i

A∗(F2,F2):

hi survives and detects Hopf map for i ≤ 3hi supports nonzero differentials for i ≥ 4

I h2j ∈ Ext2,2j+1

A∗(F2,F2):

h2j survives and detects Kervaire class θj for j ≤ 5

h2j supports differential for j ≥ 7

the fate of h26 is unknown

I gk = 〈h2k+2, hk−1, hk , hk+1〉 ∈ Ext4,2k+2+2k+3

A∗(F2,F2):

g1, g2 survive and detect κ, κ2 in stems 20, 44

g3 supports a nonzero differential in stem 92the fate of gk for k ≥ 4 is unknown

Some classes on the Adams E2-page

I hi ∈ Ext1,2i

A∗(F2,F2):

hi survives and detects Hopf map for i ≤ 3hi supports nonzero differentials for i ≥ 4

I h2j ∈ Ext2,2j+1

A∗(F2,F2):

h2j survives and detects Kervaire class θj for j ≤ 5

h2j supports differential for j ≥ 7

the fate of h26 is unknown

I gk = 〈h2k+2, hk−1, hk , hk+1〉 ∈ Ext4,2k+2+2k+3

A∗(F2,F2):

g1, g2 survive and detect κ, κ2 in stems 20, 44g3 supports a nonzero differential in stem 92

the fate of gk for k ≥ 4 is unknown

Some classes on the Adams E2-page

I hi ∈ Ext1,2i

A∗(F2,F2):

hi survives and detects Hopf map for i ≤ 3hi supports nonzero differentials for i ≥ 4

I h2j ∈ Ext2,2j+1

A∗(F2,F2):

h2j survives and detects Kervaire class θj for j ≤ 5

h2j supports differential for j ≥ 7

the fate of h26 is unknown

I gk = 〈h2k+2, hk−1, hk , hk+1〉 ∈ Ext4,2k+2+2k+3

A∗(F2,F2):

g1, g2 survive and detect κ, κ2 in stems 20, 44g3 supports a nonzero differential in stem 92the fate of gk for k ≥ 4 is unknown

0 4 8 12 16 20 24 28 32 36 40

0

4

8

12

η

v1aσ

η2

v21a2σ

0 4 8 12 16 20 24 28 32 36 40 44 48 52

0

4

8

12

16

20

24

28

32

36

40

v2a3σ

ν

v22a6σ

ν2 v42u42σa

4σ

κ

0 4 8 12 16 20 24 28 32 36 40 44 48 52

0

4

8

12

16

20

24

28

32

36

40

v3a7σ

σ

v23a14σ

σ2 v43u82σa

12σ

κ2

Real Orientation

central C2 ⊂ Sn: acts on En∗ by [−1].

MUR En

MU∗ En∗

C2

?

C2

C2 C2

Real Orientation

central C2 ⊂ Sn: acts on En∗ by [−1].

MUR En

MU∗ En∗

C2

?

C2

C2 C2

Real Orientation

central C2 ⊂ Sn: acts on En∗ by [−1].

MUR En

MU∗ En∗

C2

?

C2

C2 C2

Real Orientation

central C2 ⊂ Sn: acts on En∗ by [−1].

MUR En

MU∗ En∗

C2

?

C2

C2 C2

Can we lift it?

Theorem (Hahn–S.)

The Morava E -theory is Real oriented: it receives a C2-equivariantmap

MUR −→ En

from the Real bordism spectrum MUR.

BPR · · · BPR〈3〉 BPR〈2〉 BPR〈1〉

E3 E2 E1

C2 C2 C2

BPC2R · · · BPR〈3〉C2 BPR〈2〉C2 BPR〈1〉C2

EhC23 EhC2

2 EhC21

Detection theorems for BPR〈n〉C2

=⇒ Detection theorems for EhC2n .

BPC2R · · · BPR〈3〉C2 BPR〈2〉C2 BPR〈1〉C2

EhC23 EhC2

2 EhC21

Detection theorems for BPR〈n〉C2

=⇒ Detection theorems for EhC2n .

Theorem (Li–S.–Wang–Xu, Hahn–S.)

EhC2n detects the first n elements of the Hopf- and Kervaire-family,

and the first (n − 1) elements of the κ-family.

The Hurewicz images of EhC2n and BPR〈n〉C2 are the same.

Theorem (Li–S.–Wang–Xu, Hahn–S.)

EhC2n detects the first n elements of the Hopf- and Kervaire-family,

and the first (n − 1) elements of the κ-family.

The Hurewicz images of EhC2n and BPR〈n〉C2 are the same.

What about the periodicity and the gap?

I π∗EhC2n is 2n+2-periodic.

I πiEhC2n = 0 for i = −1,−2,−3

What about the periodicity and the gap?

I π∗EhC2n is 2n+2-periodic.

I πiEhC2n = 0 for i = −1,−2,−3

What about the periodicity and the gap?

I π∗EhC2n is 2n+2-periodic.

I πiEhC2n = 0 for i = −1,−2,−3

What about bigger groups?

Theorem (Hahn–Shi)

Let G ⊂ Sn be a finite subgroup containing the central subgroupC2. There is a G -equivariant map

MU((G)) −→ En.

What about bigger groups?

Theorem (Hahn–Shi)

Let G ⊂ Sn be a finite subgroup containing the central subgroupC2. There is a G -equivariant map

MU((G)) −→ En.

For G = C4:

BP((C4)) BP((C4))〈3〉 BP((C4))〈2〉 BP((C4))〈1〉

E6 E4 E2

C4 C4 C4

For G = C2m :

BP((C2m )) BP((C2m ))〈3〉 BP((C2m ))〈2〉 BP((C2m ))〈1〉

E3·2m−1 E2·2m−1 E2m−1

C2m C2m C2m

Periodicity and Gap Theorems for Morava E -theories

BP((C2m )) BP((C2m ))〈n〉

En·2m−1

C2m

Techniques developed by Hill, Hopkins, and Ravenel show that

I π∗EhC2m

n·2m−1 is periodic with period 2n·2m−1+m+1.

I πiEhC2m

n·2m−1 = 0 for i = −1,−2,−3.

Periodicity and Gap Theorems for Morava E -theories

BP((C2m )) BP((C2m ))〈n〉

En·2m−1

C2m

Techniques developed by Hill, Hopkins, and Ravenel show that

I π∗EhC2m

n·2m−1 is periodic with period 2n·2m−1+m+1.

I πiEhC2m

n·2m−1 = 0 for i = −1,−2,−3.

Periodicity and Gap Theorems for Morava E -theories

BP((C2m )) BP((C2m ))〈n〉

En·2m−1

C2m

Techniques developed by Hill, Hopkins, and Ravenel show that

I π∗EhC2m

n·2m−1 is periodic with period 2n·2m−1+m+1.

I πiEhC2m

n·2m−1 = 0 for i = −1,−2,−3.

Periodicity and Gap Theorems for Morava E -theories

BP((C2m )) BP((C2m ))〈n〉

En·2m−1

C2m

Techniques developed by Hill, Hopkins, and Ravenel show that

I π∗EhC2m

n·2m−1 is periodic with period 2n·2m−1+m+1.

I πiEhC2m

n·2m−1 = 0 for i = −1,−2,−3.

BP((C4)) BP((C4))〈3〉 BP((C4))〈2〉 BP((C4))〈1〉

E6 E4 E2

C4 C4 C4

SliceSS(BP ((C4))〈1〉)

0 4 8 12 16 20 24 28 32 36 40 44

0

4

8

12

16

20

24

28

32

36

SliceSS(BP ((C4))〈1〉)

0 4 8 12 16 20 24 28 32 36 40 44

0

4

8

12

16

20

24

28

32

36

Three periodicities

I RO(C4) = Z⊕ Z⊕ Z, generated by 1, σ, and λ

I D−1BP((C4))〈1〉 has three periodicities:I Sρ4 ∧ D−1BP((C4))〈1〉 ' D−1BP((C4))〈1〉I S4−4σ ∧ D−1BP((C4))〈1〉 ' D−1BP((C4))〈1〉I S8+8σ−8λ ∧ D−1BP((C4))〈1〉 ' D−1BP((C4))〈1〉

I Together, they imply the 32-periodicity!8ρ4 + 4(4− 4σ) + (8 + 8σ − 8λ)= 8(1 + σ + λ) + 4(4− 4σ) + (8 + 8σ − 8λ)= 32

Three periodicities

I RO(C4) = Z⊕ Z⊕ Z, generated by 1, σ, and λ

I D−1BP((C4))〈1〉 has three periodicities:

I Sρ4 ∧ D−1BP((C4))〈1〉 ' D−1BP((C4))〈1〉I S4−4σ ∧ D−1BP((C4))〈1〉 ' D−1BP((C4))〈1〉I S8+8σ−8λ ∧ D−1BP((C4))〈1〉 ' D−1BP((C4))〈1〉

I Together, they imply the 32-periodicity!8ρ4 + 4(4− 4σ) + (8 + 8σ − 8λ)= 8(1 + σ + λ) + 4(4− 4σ) + (8 + 8σ − 8λ)= 32

Three periodicities

I RO(C4) = Z⊕ Z⊕ Z, generated by 1, σ, and λ

I D−1BP((C4))〈1〉 has three periodicities:I Sρ4 ∧ D−1BP((C4))〈1〉 ' D−1BP((C4))〈1〉

I S4−4σ ∧ D−1BP((C4))〈1〉 ' D−1BP((C4))〈1〉I S8+8σ−8λ ∧ D−1BP((C4))〈1〉 ' D−1BP((C4))〈1〉

I Together, they imply the 32-periodicity!8ρ4 + 4(4− 4σ) + (8 + 8σ − 8λ)= 8(1 + σ + λ) + 4(4− 4σ) + (8 + 8σ − 8λ)= 32

Three periodicities

I RO(C4) = Z⊕ Z⊕ Z, generated by 1, σ, and λ

I D−1BP((C4))〈1〉 has three periodicities:I Sρ4 ∧ D−1BP((C4))〈1〉 ' D−1BP((C4))〈1〉I S4−4σ ∧ D−1BP((C4))〈1〉 ' D−1BP((C4))〈1〉

I S8+8σ−8λ ∧ D−1BP((C4))〈1〉 ' D−1BP((C4))〈1〉I Together, they imply the 32-periodicity!

8ρ4 + 4(4− 4σ) + (8 + 8σ − 8λ)= 8(1 + σ + λ) + 4(4− 4σ) + (8 + 8σ − 8λ)= 32

Three periodicities

I RO(C4) = Z⊕ Z⊕ Z, generated by 1, σ, and λ

I D−1BP((C4))〈1〉 has three periodicities:I Sρ4 ∧ D−1BP((C4))〈1〉 ' D−1BP((C4))〈1〉I S4−4σ ∧ D−1BP((C4))〈1〉 ' D−1BP((C4))〈1〉I S8+8σ−8λ ∧ D−1BP((C4))〈1〉 ' D−1BP((C4))〈1〉

I Together, they imply the 32-periodicity!8ρ4 + 4(4− 4σ) + (8 + 8σ − 8λ)= 8(1 + σ + λ) + 4(4− 4σ) + (8 + 8σ − 8λ)= 32

Three periodicities

I RO(C4) = Z⊕ Z⊕ Z, generated by 1, σ, and λ

I D−1BP((C4))〈1〉 has three periodicities:I Sρ4 ∧ D−1BP((C4))〈1〉 ' D−1BP((C4))〈1〉I S4−4σ ∧ D−1BP((C4))〈1〉 ' D−1BP((C4))〈1〉I S8+8σ−8λ ∧ D−1BP((C4))〈1〉 ' D−1BP((C4))〈1〉

I Together, they imply the 32-periodicity!8ρ4 + 4(4− 4σ) + (8 + 8σ − 8λ)= 8(1 + σ + λ) + 4(4− 4σ) + (8 + 8σ − 8λ)= 32

Hurewicz images

BPR −→ i∗C2BP((C4)) −→ i∗C2

BP((C4))〈1〉

=⇒ η, η2, ν, ν2, κ are detected in πC2∗ BP((C4))〈1〉

πC4∗ BP((C4))〈1〉 πC2

∗ BP((C4))〈1〉

π∗S

res

=⇒ These elements are also detected in πC4∗ BP((C4))〈1〉

Hurewicz images

BPR −→ i∗C2BP((C4)) −→ i∗C2

BP((C4))〈1〉

=⇒ η, η2, ν, ν2, κ are detected in πC2∗ BP((C4))〈1〉

πC4∗ BP((C4))〈1〉 πC2

∗ BP((C4))〈1〉

π∗S

res

=⇒ These elements are also detected in πC4∗ BP((C4))〈1〉

Hurewicz images

BPR −→ i∗C2BP((C4)) −→ i∗C2

BP((C4))〈1〉

=⇒ η, η2, ν, ν2, κ are detected in πC2∗ BP((C4))〈1〉

πC4∗ BP((C4))〈1〉 πC2

∗ BP((C4))〈1〉

π∗S

res

=⇒ These elements are also detected in πC4∗ BP((C4))〈1〉

Hurewicz images

BPR −→ i∗C2BP((C4)) −→ i∗C2

BP((C4))〈1〉

=⇒ η, η2, ν, ν2, κ are detected in πC2∗ BP((C4))〈1〉

πC4∗ BP((C4))〈1〉 πC2

∗ BP((C4))〈1〉

π∗S

res

=⇒ These elements are also detected in πC4∗ BP((C4))〈1〉

SliceSS(BP ((C4))〈1〉)

0 4 8 12 16 20 24 28 32 36 40 44

0

4

8

12

16

20

24

28

32

36

ηη2

ε

κ,2κ

εκ=κ2

κ2

ν2 κ

εκ

ν

ν3=ηε

2ν

SliceSS(BPR〈2〉)

0 4 8 12 16 20 24 28 32 36 40 44 48 52

0

4

8

12

16

20

24

28

32

36

40

v2a3σ

ν

v22a6σ

ν2 v42u42σa

4σ

κ

Stabilization of Filtration

For ν:

πC2m∗ BP((C2m )) C2 C4 C8 C16

Filtration 3 1 1 1

Order 2 4 4 4

For θn:

πC2m∗ BP((C2m )) C2 C4 C8 C16

η2 2 2 2 2

ν2 6 2 2 2

σ2 14 10 2 2

θ4 30 18 2 2

Stabilization of Filtration

For ν:

πC2m∗ BP((C2m )) C2 C4 C8 C16

Filtration 3 1 1 1

Order 2 4 4 4

For θn:

πC2m∗ BP((C2m )) C2 C4 C8 C16

η2 2 2 2 2

ν2 6 2 2 2

σ2 14 10 2 2

θ4 30 18 2 2

Hill’s Detection Tower...

π∗(MU((C2m )))C2m

...

π∗S π∗(MU((C8)))C8

π∗(MU((C4)))C4

π∗(MUR)C2

Conjecture (Hill)

1. As m increases, the filtration of a spherical class detected inπ∗(MU((C2m )))C2m decreases and eventually stabilizes to itsAdams–Novikov filtration.

2. When this class first moves into its stable filtration, itachieves its maximal order that is detected byπ∗(MU((C2m )))C2m for all m.

Question (Hill)

What is the Hurewicz image of lim←−π∗(MU((C2m )))C2m ?

Conjecture (Hill)

1. As m increases, the filtration of a spherical class detected inπ∗(MU((C2m )))C2m decreases and eventually stabilizes to itsAdams–Novikov filtration.

2. When this class first moves into its stable filtration, itachieves its maximal order that is detected byπ∗(MU((C2m )))C2m for all m.

Question (Hill)

What is the Hurewicz image of lim←−π∗(MU((C2m )))C2m ?

Conjecture (Hill)

1. As m increases, the filtration of a spherical class detected inπ∗(MU((C2m )))C2m decreases and eventually stabilizes to itsAdams–Novikov filtration.

2. When this class first moves into its stable filtration, itachieves its maximal order that is detected byπ∗(MU((C2m )))C2m for all m.

Question (Hill)

What is the Hurewicz image of lim←−π∗(MU((C2m )))C2m ?

Hill’s Detection Tower

...

π∗(MU((C2m )))C2m π∗EhC2m

2m−1n

...

π∗S π∗(MU((C8)))C8 π∗EhC84n

π∗(MU((C4)))C4 π∗EhC42n

π∗(MUR)C2 π∗EhC2n

Hill’s Detection Tower

...

π∗(MU((C2m )))C2m π∗EhC2m

2m−1n

...

π∗S π∗(MU((C8)))C8 π∗EhC84n

π∗(MU((C4)))C4 π∗EhC42n

π∗(MUR)C2 π∗EhC2n

Slice SS vs. Homotopy Fixed Point SS

SliceSS(X ) HFPSS(X )

π∗XG π∗X

hG

This is an isomorphism under the line of slope 1

Slice SS vs. Homotopy Fixed Point SS

SliceSS(X ) HFPSS(X )

π∗XG π∗X

hG

This is an isomorphism under the line of slope 1

HFPSS(E hC4

2 )

SliceSS(E hC4

2 )

Advantages of the Slice SS

I Gap Theorem is immediate at the E2-page.

I There is a region consisting of only regular G -slice cells. It’ssparse.

I Hill, Hopkins, and Ravenel proved all the differentials in thisregion (the Slice Differential Theorem).

I These differentials imply the Periodicity Theorem.

I Slice SS + isomorphism + Periodicity Theoremrecovers

=⇒ HFPSS

Advantages of the Slice SS

I Gap Theorem is immediate at the E2-page.

I There is a region consisting of only regular G -slice cells. It’ssparse.

I Hill, Hopkins, and Ravenel proved all the differentials in thisregion (the Slice Differential Theorem).

I These differentials imply the Periodicity Theorem.

I Slice SS + isomorphism + Periodicity Theoremrecovers

=⇒ HFPSS

Advantages of the Slice SS

I Gap Theorem is immediate at the E2-page.

I There is a region consisting of only regular G -slice cells. It’ssparse.

I Hill, Hopkins, and Ravenel proved all the differentials in thisregion (the Slice Differential Theorem).

I These differentials imply the Periodicity Theorem.

I Slice SS + isomorphism + Periodicity Theoremrecovers

=⇒ HFPSS

Advantages of the Slice SS

I Gap Theorem is immediate at the E2-page.

I There is a region consisting of only regular G -slice cells. It’ssparse.

I Hill, Hopkins, and Ravenel proved all the differentials in thisregion (the Slice Differential Theorem).

I These differentials imply the Periodicity Theorem.

I Slice SS + isomorphism + Periodicity Theoremrecovers

=⇒ HFPSS

Advantages of the Slice SS

I Gap Theorem is immediate at the E2-page.

I There is a region consisting of only regular G -slice cells. It’ssparse.

I Hill, Hopkins, and Ravenel proved all the differentials in thisregion (the Slice Differential Theorem).

I These differentials imply the Periodicity Theorem.

I Slice SS + isomorphism + Periodicity Theoremrecovers

=⇒ HFPSS

Computing Differentials

MU((C4)): commutative C4-spectrum

πC4F MU((C4)) πC2

F MU((C4)).

res

tr

norm

Computing Differentials

C4- SliceSS(MU((C4))) C2- SliceSS(MU((C4))).

res

tr

norm

I Restriction and transfer are maps of spectral sequences.

I Norm “stretches out” differentials.

Computing Differentials

C4- SliceSS(MU((C4))) C2- SliceSS(MU((C4))).

res

tr

norm

I Restriction and transfer are maps of spectral sequences.

I Norm “stretches out” differentials.

Computing Differentials

C4- SliceSS(MU((C4))) C2- SliceSS(MU((C4))).

res

tr

norm

I Restriction and transfer are maps of spectral sequences.

I Norm “stretches out” differentials.

Theorem (Hill–Hopkins–Ravenel)

Let dr (x) = y be a dr -differential in C2-SliceSS(MU((C4))). If bothaσN

C4C2x and NC4

C2y survive to the E2r−1-page in

C4-SliceSS(MU((C4))), then

d2r−1(aσNC4C2x) = NC4

C2y .

Restriction

d31uλu2σaσ r41γ r31 a

7σ2

d21u2λu2σ r21γ r21u4σ2

C4-SliceSS C2-SliceSS

d5

res

d7

Transfer

d31s1u3σa

3λaσ2 r41γ r

31 a

7σ2

2d21u2λu2σ r21γ r21u4σ2

C4-SliceSS C2-SliceSS

tr

d7

tr

d7

Norm

C2-SliceSS: d7(u4σ2) = r21γ r1a7σ2

C4-SliceSS: d13(u4λaσ) = d31a7λ

apply norm

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SliceSS(BP ((C4))〈2〉)

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BP((C4)) BP((C4))〈3〉 BP((C4))〈2〉 BP((C4))〈1〉

E6 E4 E2

C4 C4 C4

SliceSS(BP ((C4))〈2〉) : d13

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SliceSS(BP ((C4))〈2〉) : d15

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SliceSS(BP ((C4))〈2〉) : d19

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SliceSS(BP ((C4))〈2〉) : d21

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SliceSS(BP ((C4))〈2〉) : d23

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SliceSS(BP ((C4))〈2〉) : d27

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SliceSS(BP ((C4))〈2〉) : d29

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SliceSS(BP ((C4))〈2〉) : d31

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SliceSS(BP ((C4))〈2〉) : d35

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SliceSS(BP ((C4))〈2〉) : d43

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SliceSS(BP ((C4))〈2〉) : d51

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SliceSS(BP ((C4))〈2〉) : d53

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SliceSS(BP ((C4))〈2〉) : d55

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SliceSS(BP ((C4))〈2〉) : d59

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SliceSS(BP ((C4))〈2〉) : d61

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SliceSS(BP ((C4))〈2〉) : E∞

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SliceSS(BP ((C4))〈2〉): Hurewicz Images

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κ2

κ2

κ3

ηη2

ν

2ν

ν2

ε

κ

ν3=ηε

κ,2κ

εκεκ=κ2

σ

σ2

2σ

θ4

1

Periodicities

I There are three periodicities for D−1BP((C4))〈2〉:

I S3ρ4 ∧ D−1BP((C4))〈2〉 ' D−1BP((C4))〈2〉I S32+32σ−32λ ∧ D−1BP((C4))〈2〉 ' D−1BP((C4))〈2〉I S8−8σ ∧ D−1BP((C4))〈2〉 ' D−1BP((C4))〈2〉

I These periodicities imply that D−1BP((C4))〈2〉 is 384-periodic!32 · (3ρ4) + 3 · (32 + 32σ − 32λ) + 24 · (8− 8σ)= 32 · (3 + 3σ + 3λ) + 3 · (32 + 32σ − 32λ) + 24 · (8− 8σ)= 384

Periodicities

I There are three periodicities for D−1BP((C4))〈2〉:I S3ρ4 ∧ D−1BP((C4))〈2〉 ' D−1BP((C4))〈2〉

I S32+32σ−32λ ∧ D−1BP((C4))〈2〉 ' D−1BP((C4))〈2〉I S8−8σ ∧ D−1BP((C4))〈2〉 ' D−1BP((C4))〈2〉

I These periodicities imply that D−1BP((C4))〈2〉 is 384-periodic!32 · (3ρ4) + 3 · (32 + 32σ − 32λ) + 24 · (8− 8σ)= 32 · (3 + 3σ + 3λ) + 3 · (32 + 32σ − 32λ) + 24 · (8− 8σ)= 384

Periodicities

I There are three periodicities for D−1BP((C4))〈2〉:I S3ρ4 ∧ D−1BP((C4))〈2〉 ' D−1BP((C4))〈2〉I S32+32σ−32λ ∧ D−1BP((C4))〈2〉 ' D−1BP((C4))〈2〉

I S8−8σ ∧ D−1BP((C4))〈2〉 ' D−1BP((C4))〈2〉I These periodicities imply that D−1BP((C4))〈2〉 is 384-periodic!

32 · (3ρ4) + 3 · (32 + 32σ − 32λ) + 24 · (8− 8σ)= 32 · (3 + 3σ + 3λ) + 3 · (32 + 32σ − 32λ) + 24 · (8− 8σ)= 384

Periodicities

I There are three periodicities for D−1BP((C4))〈2〉:I S3ρ4 ∧ D−1BP((C4))〈2〉 ' D−1BP((C4))〈2〉I S32+32σ−32λ ∧ D−1BP((C4))〈2〉 ' D−1BP((C4))〈2〉I S8−8σ ∧ D−1BP((C4))〈2〉 ' D−1BP((C4))〈2〉

I These periodicities imply that D−1BP((C4))〈2〉 is 384-periodic!32 · (3ρ4) + 3 · (32 + 32σ − 32λ) + 24 · (8− 8σ)= 32 · (3 + 3σ + 3λ) + 3 · (32 + 32σ − 32λ) + 24 · (8− 8σ)= 384

Periodicities

I There are three periodicities for D−1BP((C4))〈2〉:I S3ρ4 ∧ D−1BP((C4))〈2〉 ' D−1BP((C4))〈2〉I S32+32σ−32λ ∧ D−1BP((C4))〈2〉 ' D−1BP((C4))〈2〉I S8−8σ ∧ D−1BP((C4))〈2〉 ' D−1BP((C4))〈2〉

I These periodicities imply that D−1BP((C4))〈2〉 is 384-periodic!32 · (3ρ4) + 3 · (32 + 32σ − 32λ) + 24 · (8− 8σ)= 32 · (3 + 3σ + 3λ) + 3 · (32 + 32σ − 32λ) + 24 · (8− 8σ)= 384

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