Post on 11-Oct-2020
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〉)
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132 136 140 144 148 152 156 160 164 168 172 176 180 184 188 192 196 200 204 208 212 216 220 224 228 232 236 240 244 248 252 256 260 264 268 272 276 280 284 288 292 296 300 304 308 312 316 320 324 328 332 336 340 344 348 352 356 360 364 368 372 376 380 384 388 392 396 400
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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
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132 136 140 144 148 152 156 160 164 168 172 176 180 184 188 192 196 200 204 208 212 216 220 224 228 232 236 240 244 248 252 256 260 264 268 272 276 280 284 288 292 296 300 304 308 312 316 320 324 328 332 336 340 344 348 352 356 360 364 368 372 376 380 384 388 392 396 400
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SliceSS(BP ((C4))〈2〉) : d15
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132 136 140 144 148 152 156 160 164 168 172 176 180 184 188 192 196 200 204 208 212 216 220 224 228 232 236 240 244 248 252 256 260 264 268 272 276 280 284 288 292 296 300 304 308 312 316 320 324 328 332 336 340 344 348 352 356 360 364 368 372 376 380 384 388 392 396 400
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SliceSS(BP ((C4))〈2〉) : d19
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132 136 140 144 148 152 156 160 164 168 172 176 180 184 188 192 196 200 204 208 212 216 220 224 228 232 236 240 244 248 252 256 260 264 268 272 276 280 284 288 292 296 300 304 308 312 316 320 324 328 332 336 340 344 348 352 356 360 364 368 372 376 380 384 388 392 396 400
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SliceSS(BP ((C4))〈2〉) : d21
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132 136 140 144 148 152 156 160 164 168 172 176 180 184 188 192 196 200 204 208 212 216 220 224 228 232 236 240 244 248 252 256 260 264 268 272 276 280 284 288 292 296 300 304 308 312 316 320 324 328 332 336 340 344 348 352 356 360 364 368 372 376 380 384 388 392 396 400
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SliceSS(BP ((C4))〈2〉) : d23
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132 136 140 144 148 152 156 160 164 168 172 176 180 184 188 192 196 200 204 208 212 216 220 224 228 232 236 240 244 248 252 256 260 264 268 272 276 280 284 288 292 296 300 304 308 312 316 320 324 328 332 336 340 344 348 352 356 360 364 368 372 376 380 384 388 392 396 400
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SliceSS(BP ((C4))〈2〉) : d27
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132 136 140 144 148 152 156 160 164 168 172 176 180 184 188 192 196 200 204 208 212 216 220 224 228 232 236 240 244 248 252 256 260 264 268 272 276 280 284 288 292 296 300 304 308 312 316 320 324 328 332 336 340 344 348 352 356 360 364 368 372 376 380 384 388 392 396 400
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SliceSS(BP ((C4))〈2〉) : d29
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132 136 140 144 148 152 156 160 164 168 172 176 180 184 188 192 196 200 204 208 212 216 220 224 228 232 236 240 244 248 252 256 260 264 268 272 276 280 284 288 292 296 300 304 308 312 316 320 324 328 332 336 340 344 348 352 356 360 364 368 372 376 380 384 388 392 396 400
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SliceSS(BP ((C4))〈2〉) : d31
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132 136 140 144 148 152 156 160 164 168 172 176 180 184 188 192 196 200 204 208 212 216 220 224 228 232 236 240 244 248 252 256 260 264 268 272 276 280 284 288 292 296 300 304 308 312 316 320 324 328 332 336 340 344 348 352 356 360 364 368 372 376 380 384 388 392 396 400
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SliceSS(BP ((C4))〈2〉) : d35
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132 136 140 144 148 152 156 160 164 168 172 176 180 184 188 192 196 200 204 208 212 216 220 224 228 232 236 240 244 248 252 256 260 264 268 272 276 280 284 288 292 296 300 304 308 312 316 320 324 328 332 336 340 344 348 352 356 360 364 368 372 376 380 384 388 392 396 400
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SliceSS(BP ((C4))〈2〉) : d43
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132 136 140 144 148 152 156 160 164 168 172 176 180 184 188 192 196 200 204 208 212 216 220 224 228 232 236 240 244 248 252 256 260 264 268 272 276 280 284 288 292 296 300 304 308 312 316 320 324 328 332 336 340 344 348 352 356 360 364 368 372 376 380 384 388 392 396 400
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SliceSS(BP ((C4))〈2〉) : d51
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132 136 140 144 148 152 156 160 164 168 172 176 180 184 188 192 196 200 204 208 212 216 220 224 228 232 236 240 244 248 252 256 260 264 268 272 276 280 284 288 292 296 300 304 308 312 316 320 324 328 332 336 340 344 348 352 356 360 364 368 372 376 380 384 388 392 396 400
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SliceSS(BP ((C4))〈2〉) : d53
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132 136 140 144 148 152 156 160 164 168 172 176 180 184 188 192 196 200 204 208 212 216 220 224 228 232 236 240 244 248 252 256 260 264 268 272 276 280 284 288 292 296 300 304 308 312 316 320 324 328 332 336 340 344 348 352 356 360 364 368 372 376 380 384 388 392 396 400
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SliceSS(BP ((C4))〈2〉) : d55
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132 136 140 144 148 152 156 160 164 168 172 176 180 184 188 192 196 200 204 208 212 216 220 224 228 232 236 240 244 248 252 256 260 264 268 272 276 280 284 288 292 296 300 304 308 312 316 320 324 328 332 336 340 344 348 352 356 360 364 368 372 376 380 384 388 392 396 400
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SliceSS(BP ((C4))〈2〉) : d59
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132 136 140 144 148 152 156 160 164 168 172 176 180 184 188 192 196 200 204 208 212 216 220 224 228 232 236 240 244 248 252 256 260 264 268 272 276 280 284 288 292 296 300 304 308 312 316 320 324 328 332 336 340 344 348 352 356 360 364 368 372 376 380 384 388 392 396 400
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SliceSS(BP ((C4))〈2〉) : d61
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132 136 140 144 148 152 156 160 164 168 172 176 180 184 188 192 196 200 204 208 212 216 220 224 228 232 236 240 244 248 252 256 260 264 268 272 276 280 284 288 292 296 300 304 308 312 316 320 324 328 332 336 340 344 348 352 356 360 364 368 372 376 380 384 388 392 396 400
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SliceSS(BP ((C4))〈2〉) : E∞
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132 136 140 144 148 152 156 160 164 168 172 176 180 184 188 192 196 200 204 208 212 216 220 224 228 232 236 240 244 248 252 256 260 264 268 272 276 280 284 288 292 296 300 304 308 312 316 320 324 328 332 336 340 344 348 352 356 360 364 368 372 376 380 384 388 392 396 400
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SliceSS(BP ((C4))〈2〉): Hurewicz Images
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132 136 140 144 148 152 156 160 164 168 172 176 180 184 188 192 196 200 204 208 212 216 220 224 228 232 236 240 244 248 252 256 260 264 268 272 276 280 284 288 292 296 300 304 308 312 316 320 324 328 332 336 340 344 348 352 356 360 364 368 372 376 380 384 388 392 396 400
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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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