the slice spectral sequence of a height @let@token 4 theory · the slice spectral sequence of a...
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![Page 1: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/1.jpg)
The Slice Spectral Sequence ofa Height 4 Theory
XiaoLin Danny Shi
(Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)
Harvard
June 8, 2018
![Page 2: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/2.jpg)
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![Page 3: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/3.jpg)
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![Page 4: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/4.jpg)
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
![Page 5: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/5.jpg)
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
![Page 6: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/6.jpg)
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
![Page 7: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/7.jpg)
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
![Page 8: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/8.jpg)
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
![Page 9: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/9.jpg)
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
![Page 10: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/10.jpg)
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
![Page 11: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/11.jpg)
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
![Page 12: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/12.jpg)
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
![Page 13: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/13.jpg)
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
![Page 14: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/14.jpg)
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
![Page 15: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/15.jpg)
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
![Page 16: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/16.jpg)
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
![Page 17: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/17.jpg)
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
![Page 18: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/18.jpg)
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
![Page 19: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/19.jpg)
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
![Page 20: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/20.jpg)
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
![Page 21: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/21.jpg)
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
![Page 22: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/22.jpg)
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
![Page 23: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/23.jpg)
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
![Page 24: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/24.jpg)
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
![Page 25: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/25.jpg)
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
![Page 26: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/26.jpg)
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!
![Page 27: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/27.jpg)
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!
![Page 28: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/28.jpg)
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!
![Page 29: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/29.jpg)
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!
![Page 30: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/30.jpg)
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!
![Page 31: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/31.jpg)
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
![Page 32: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/32.jpg)
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
![Page 33: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/33.jpg)
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!
![Page 34: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/34.jpg)
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 )
![Page 35: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/35.jpg)
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 )
![Page 36: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/36.jpg)
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 )
![Page 37: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/37.jpg)
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 )
![Page 38: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/38.jpg)
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 )
![Page 39: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/39.jpg)
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 )
![Page 40: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/40.jpg)
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
![Page 41: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/41.jpg)
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
![Page 42: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/42.jpg)
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
![Page 43: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/43.jpg)
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
![Page 44: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/44.jpg)
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
![Page 45: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/45.jpg)
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
![Page 46: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/46.jpg)
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
![Page 47: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/47.jpg)
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
![Page 48: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/48.jpg)
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
![Page 49: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/49.jpg)
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
![Page 50: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/50.jpg)
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
![Page 51: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/51.jpg)
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 ))
![Page 52: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/52.jpg)
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 ))
![Page 53: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/53.jpg)
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 ))
![Page 54: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/54.jpg)
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 ))
![Page 55: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/55.jpg)
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 ))
![Page 56: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/56.jpg)
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 ))
![Page 57: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/57.jpg)
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.
![Page 58: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/58.jpg)
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.
![Page 59: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/59.jpg)
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.
![Page 60: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/60.jpg)
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.
![Page 61: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/61.jpg)
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.
![Page 62: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/62.jpg)
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.
![Page 63: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/63.jpg)
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.
![Page 64: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/64.jpg)
Baby Ω
Gap Theorem
Periodicity Theorem
![Page 65: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/65.jpg)
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 !
![Page 66: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/66.jpg)
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 !
![Page 67: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/67.jpg)
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 !
![Page 68: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/68.jpg)
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 !
![Page 69: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/69.jpg)
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 !
![Page 70: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/70.jpg)
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
![Page 71: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/71.jpg)
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
![Page 72: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/72.jpg)
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
![Page 73: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/73.jpg)
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
![Page 74: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/74.jpg)
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
![Page 75: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/75.jpg)
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
![Page 76: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/76.jpg)
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
![Page 77: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/77.jpg)
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
![Page 78: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/78.jpg)
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
![Page 79: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/79.jpg)
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
![Page 80: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/80.jpg)
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
![Page 81: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/81.jpg)
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
![Page 82: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/82.jpg)
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
![Page 83: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/83.jpg)
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
![Page 84: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/84.jpg)
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
![Page 85: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/85.jpg)
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
![Page 86: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/86.jpg)
This is a spectral sequence of Mackey functors.
It is also a Mackey functor of spectral sequences!
![Page 87: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/87.jpg)
This is a spectral sequence of Mackey functors.
It is also a Mackey functor of spectral sequences!
![Page 88: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/88.jpg)
Slice tower:
X · · · PnX Pn−1X · · ·
PnnX Pn−1
n−1X
![Page 89: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/89.jpg)
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 )
![Page 90: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/90.jpg)
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 )
![Page 91: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/91.jpg)
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
![Page 92: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/92.jpg)
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
![Page 93: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/93.jpg)
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
![Page 94: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/94.jpg)
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
![Page 95: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/95.jpg)
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
![Page 96: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/96.jpg)
C2-SliceSS(KR): π∗KR
−16 −12 −8 −4 0 4 8 12 16
−8
−4
0
4
8
![Page 97: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/97.jpg)
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
![Page 98: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/98.jpg)
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
![Page 99: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/99.jpg)
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
![Page 100: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/100.jpg)
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
![Page 101: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/101.jpg)
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
![Page 102: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/102.jpg)
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
![Page 103: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/103.jpg)
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
![Page 104: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/104.jpg)
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
![Page 105: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/105.jpg)
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
![Page 106: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/106.jpg)
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
![Page 107: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/107.jpg)
C2-SliceSS(KR): π∗KR
−16 −12 −8 −4 0 4 8 12 16
−8
−4
0
4
8
![Page 108: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/108.jpg)
C2-SliceSS(KR): πu∗KR = π∗KU
−16 −12 −8 −4 0 4 8 12 16
−8
−4
0
4
8
![Page 109: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/109.jpg)
C2-SliceSS(KR): πC2∗ KR = π∗KO
−16 −12 −8 −4 0 4 8 12 16
−8
−4
0
4
8
![Page 110: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/110.jpg)
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
![Page 111: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/111.jpg)
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
![Page 112: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/112.jpg)
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
![Page 113: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/113.jpg)
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
![Page 114: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/114.jpg)
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
![Page 115: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/115.jpg)
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
![Page 116: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/116.jpg)
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?
![Page 117: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/117.jpg)
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?
![Page 118: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/118.jpg)
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?
![Page 119: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/119.jpg)
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?
![Page 120: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/120.jpg)
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?
![Page 121: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/121.jpg)
Hurewicz Image of BPC2
R
BPR · · · BPR〈3〉 BPR〈2〉 BPR〈1〉
![Page 122: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/122.jpg)
Hurewicz Image of BPC2
R
BPC2R · · · BPR〈3〉C2 BPR〈2〉C2 BPR〈1〉C2
![Page 123: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/123.jpg)
Hurewicz Image of BPC2
R
BPC2R · · · BPR〈3〉C2 BPR〈2〉C2 BPR〈1〉C2
S
What are the Hurewicz images?
![Page 124: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/124.jpg)
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.
![Page 125: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/125.jpg)
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.
![Page 126: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/126.jpg)
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
![Page 127: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/127.jpg)
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
![Page 128: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/128.jpg)
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
![Page 129: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/129.jpg)
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
![Page 130: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/130.jpg)
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
![Page 131: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/131.jpg)
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
![Page 132: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/132.jpg)
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
![Page 133: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/133.jpg)
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
![Page 134: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/134.jpg)
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
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0 4 8 12 16 20 24 28 32 36 40
0
4
8
12
η
v1aσ
η2
v21a2σ
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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
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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
![Page 138: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/138.jpg)
Real Orientation
central C2 ⊂ Sn: acts on En∗ by [−1].
MUR En
MU∗ En∗
C2
?
C2
C2 C2
![Page 139: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/139.jpg)
Real Orientation
central C2 ⊂ Sn: acts on En∗ by [−1].
MUR En
MU∗ En∗
C2
?
C2
C2 C2
![Page 140: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/140.jpg)
Real Orientation
central C2 ⊂ Sn: acts on En∗ by [−1].
MUR En
MU∗ En∗
C2
?
C2
C2 C2
![Page 141: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/141.jpg)
Real Orientation
central C2 ⊂ Sn: acts on En∗ by [−1].
MUR En
MU∗ En∗
C2
?
C2
C2 C2
Can we lift it?
![Page 142: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/142.jpg)
Theorem (Hahn–S.)
The Morava E -theory is Real oriented: it receives a C2-equivariantmap
MUR −→ En
from the Real bordism spectrum MUR.
![Page 143: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/143.jpg)
BPR · · · BPR〈3〉 BPR〈2〉 BPR〈1〉
E3 E2 E1
C2 C2 C2
![Page 144: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/144.jpg)
BPC2R · · · BPR〈3〉C2 BPR〈2〉C2 BPR〈1〉C2
EhC23 EhC2
2 EhC21
Detection theorems for BPR〈n〉C2
=⇒ Detection theorems for EhC2n .
![Page 145: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/145.jpg)
BPC2R · · · BPR〈3〉C2 BPR〈2〉C2 BPR〈1〉C2
EhC23 EhC2
2 EhC21
Detection theorems for BPR〈n〉C2
=⇒ Detection theorems for EhC2n .
![Page 146: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/146.jpg)
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.
![Page 147: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/147.jpg)
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.
![Page 148: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/148.jpg)
What about the periodicity and the gap?
I π∗EhC2n is 2n+2-periodic.
I πiEhC2n = 0 for i = −1,−2,−3
![Page 149: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/149.jpg)
What about the periodicity and the gap?
I π∗EhC2n is 2n+2-periodic.
I πiEhC2n = 0 for i = −1,−2,−3
![Page 150: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/150.jpg)
What about the periodicity and the gap?
I π∗EhC2n is 2n+2-periodic.
I πiEhC2n = 0 for i = −1,−2,−3
![Page 151: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/151.jpg)
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.
![Page 152: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/152.jpg)
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.
![Page 153: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/153.jpg)
For G = C4:
BP((C4)) BP((C4))〈3〉 BP((C4))〈2〉 BP((C4))〈1〉
E6 E4 E2
C4 C4 C4
![Page 154: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/154.jpg)
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
![Page 155: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/155.jpg)
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.
![Page 156: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/156.jpg)
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.
![Page 157: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/157.jpg)
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.
![Page 158: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/158.jpg)
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.
![Page 159: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/159.jpg)
BP((C4)) BP((C4))〈3〉 BP((C4))〈2〉 BP((C4))〈1〉
E6 E4 E2
C4 C4 C4
![Page 160: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/160.jpg)
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
![Page 161: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/161.jpg)
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
![Page 162: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/162.jpg)
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
![Page 163: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/163.jpg)
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
![Page 164: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/164.jpg)
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
![Page 165: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/165.jpg)
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
![Page 166: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/166.jpg)
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
![Page 167: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/167.jpg)
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
![Page 168: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/168.jpg)
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〉
![Page 169: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/169.jpg)
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〉
![Page 170: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/170.jpg)
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〉
![Page 171: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/171.jpg)
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〉
![Page 172: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/172.jpg)
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
![Page 173: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/173.jpg)
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
![Page 174: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/174.jpg)
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
![Page 175: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/175.jpg)
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
![Page 176: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/176.jpg)
Hill’s Detection Tower...
π∗(MU((C2m )))C2m
...
π∗S π∗(MU((C8)))C8
π∗(MU((C4)))C4
π∗(MUR)C2
![Page 177: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/177.jpg)
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 ?
![Page 178: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/178.jpg)
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 ?
![Page 179: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/179.jpg)
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 ?
![Page 180: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/180.jpg)
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
![Page 181: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/181.jpg)
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
![Page 182: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/182.jpg)
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
![Page 183: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/183.jpg)
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
![Page 184: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/184.jpg)
HFPSS(E hC4
2 )
![Page 185: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/185.jpg)
SliceSS(E hC4
2 )
![Page 186: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/186.jpg)
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
![Page 187: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/187.jpg)
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
![Page 188: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/188.jpg)
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
![Page 189: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/189.jpg)
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
![Page 190: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/190.jpg)
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
![Page 191: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/191.jpg)
Computing Differentials
MU((C4)): commutative C4-spectrum
πC4F MU((C4)) πC2
F MU((C4)).
res
tr
norm
![Page 192: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/192.jpg)
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.
![Page 193: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/193.jpg)
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.
![Page 194: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/194.jpg)
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.
![Page 195: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/195.jpg)
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 .
![Page 196: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/196.jpg)
Restriction
d31uλu2σaσ r41γ r31 a
7σ2
d21u2λu2σ r21γ r21u4σ2
C4-SliceSS C2-SliceSS
d5
res
d7
![Page 197: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/197.jpg)
Transfer
d31s1u3σa
3λaσ2 r41γ r
31 a
7σ2
2d21u2λu2σ r21γ r21u4σ2
C4-SliceSS C2-SliceSS
tr
d7
tr
d7
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Norm
C2-SliceSS: d7(u4σ2) = r21γ r1a7σ2
C4-SliceSS: d13(u4λaσ) = d31a7λ
apply norm
![Page 199: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/199.jpg)
0 4 8 12 16 20
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?
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〉)
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
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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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1
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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
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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
![Page 222: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/222.jpg)
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
![Page 223: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/223.jpg)
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
![Page 224: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/224.jpg)
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
![Page 225: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/225.jpg)
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
![Page 226: The Slice Spectral Sequence of a Height @let@token 4 Theory · The Slice Spectral Sequence of a Height 4 Theory XiaoLin Danny Shi (Joint with Mike Hill, Guozhen Wang, and Zhouli Xu)](https://reader035.vdocuments.site/reader035/viewer/2022071214/60431dbf3abe1830135f524e/html5/thumbnails/226.jpg)
Thank you all for coming!