Unstable Homotopy Theory from the Chromatic Point of...
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The EHP SequencePeriodic Unstable Homotopy Theory
The K(2)-local Goodwillie Tower of SpheresComputation of π∗(ΦK(2)S
3)
Unstable Homotopy Theory from the ChromaticPoint of View
Guozhen Wang
MIT
April 13, 2015
Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View
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The EHP SequencePeriodic Unstable Homotopy Theory
The K(2)-local Goodwillie Tower of SpheresComputation of π∗(ΦK(2)S
3)
Outline
1 The EHP SequenceDefinition of EHP sequenceEHP sequence for p = 3Exponent of unstable homotopy groups
2 Periodic Unstable Homotopy TheoryPeriodic unstable homotopy groupsBousfield-Kuhn functorv1-periodic unstable homotopy groupsBounded torsion phenomenon in chromatic level 2
3 The K (2)-local Goodwillie Tower of SpheresThe Goodwillie tower of the identityThe Goodwillie derivatives of spheresGoodwillie differentials on En-cohomology
4 Computation of π∗(ΦK(2)S3)
Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View
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The EHP SequencePeriodic Unstable Homotopy Theory
The K(2)-local Goodwillie Tower of SpheresComputation of π∗(ΦK(2)S
3)
Definition of EHP sequenceEHP sequence for p = 3Exponent of unstable homotopy groups
Section 1
The EHP Sequence
Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View
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The EHP SequencePeriodic Unstable Homotopy Theory
The K(2)-local Goodwillie Tower of SpheresComputation of π∗(ΦK(2)S
3)
Definition of EHP sequenceEHP sequence for p = 3Exponent of unstable homotopy groups
The Hopf invariant
Theorem (James Splitting)
Let X be a connected space. Then there is a homotopyequivalence ΣΩΣX = ∨ΣX∧i .
Definition (Hopf invariant)
The Hopf map H : ΩΣX → ΩΣX∧p at prime p is defined to be theadjoint of the projection map ΣΩΣX+
∼= ∨ΣX∧i → ΣX∧p.
Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View
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The EHP SequencePeriodic Unstable Homotopy Theory
The K(2)-local Goodwillie Tower of SpheresComputation of π∗(ΦK(2)S
3)
Definition of EHP sequenceEHP sequence for p = 3Exponent of unstable homotopy groups
The EHP sequence
Theorem (James)
We have a 2-local fiber sequence:
Sk E−→ ΩSk+1 H−→ ΩS2k+1
Theorem (Toda)
At an odd prime p, we have fiber sequences:
ˆS2k E−→ ΩS2k+1 H−→ ΩS2pk+1
S2k−1 E−→ Ω ˆS2k H−→ ΩS2kp−1
where ˆS2k is the (2kp − 1)-skeleton of ΩS2k+1.
Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View
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The EHP SequencePeriodic Unstable Homotopy Theory
The K(2)-local Goodwillie Tower of SpheresComputation of π∗(ΦK(2)S
3)
Definition of EHP sequenceEHP sequence for p = 3Exponent of unstable homotopy groups
EHP sequence for p = 3
0 1 2 3 4 5 6 7 8 9 10 11 121 α1 α2 β1 α3/2
1 * * * * * * * * * * * *1 * * α1 * * * α2 * *
1 * * α1 * * * α2 *1 * * α1 * *
1 * * α1 *1 *
1
Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View
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The EHP SequencePeriodic Unstable Homotopy Theory
The K(2)-local Goodwillie Tower of SpheresComputation of π∗(ΦK(2)S
3)
Definition of EHP sequenceEHP sequence for p = 3Exponent of unstable homotopy groups
EHP sequence for p = 3
13 14 15 16 17 18 19 20 21 22 23α1β1 α4 α5 β1^2 α1β1^2 ; α6/2
* * * * * * * * * * *β1~ α3/2 * α1β1 μ[α2] α4 * * μ[α3/2] α5 β1^2
* β1 α3/2 * α1β1 μ[α1] α4 * * μ[α2] α5* α2 * * β1 α3/2 * α1β1 * α4 ** * α2 * * β1 α3/2 * α1β1 * α4* α1 * * * α2 * * β1 α3/2 ** * α1 * * * α2 * * β1 α3/2
1 * * α1 * * * α2 *1 * * α1 * * * α2
1 * * α1 *1 * * α1
1
Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View
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The EHP SequencePeriodic Unstable Homotopy Theory
The K(2)-local Goodwillie Tower of SpheresComputation of π∗(ΦK(2)S
3)
Definition of EHP sequenceEHP sequence for p = 3Exponent of unstable homotopy groups
p-exponent of unstable homotopy groups
Theorem (James, Toda)
1 The d1-differential on odd rows of the EHP spectral sequenceis the multiplication by p map.
2 The p-component of π∗S2k+1 is annialated by p2k .
Theorem (Cohen-Moore-Neisendorfer)
At an odd prime p,
1 The multiplication by p map on the fiber of double suspensionS2k−1 → Ω2S2k+1 is zero.
2 The p-component of π∗S2k+1 is annihilated by pk .
Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View
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The EHP SequencePeriodic Unstable Homotopy Theory
The K(2)-local Goodwillie Tower of SpheresComputation of π∗(ΦK(2)S
3)
Periodic unstable homotopy groupsBousfield-Kuhn functorv1-periodic unstable homotopy groupsBounded torsion phenomenon in chromatic level 2
Section 2
Periodic Unstable Homotopy Theory
Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View
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The EHP SequencePeriodic Unstable Homotopy Theory
The K(2)-local Goodwillie Tower of SpheresComputation of π∗(ΦK(2)S
3)
Periodic unstable homotopy groupsBousfield-Kuhn functorv1-periodic unstable homotopy groupsBounded torsion phenomenon in chromatic level 2
Type n complex
Definition
A finite CW -complex W is type n if K (h)∗W = 0 for h < n, and
K (n)∗W is nontrivial.
Theorem (Hopkins-Smith)
For a type n complex W , there exist positive integers t,N and map
v tn : ΣN+t|vn|W → ΣNW
such that v tn induces multiplication by v t
n on K (n)-homology.
Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View
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The EHP SequencePeriodic Unstable Homotopy Theory
The K(2)-local Goodwillie Tower of SpheresComputation of π∗(ΦK(2)S
3)
Periodic unstable homotopy groupsBousfield-Kuhn functorv1-periodic unstable homotopy groupsBounded torsion phenomenon in chromatic level 2
Periodic homotopy groups
Definition
Let X be a space. The homotopy groups of X with coefficients inW is defined by
πi (X ; W ) = [ΣiW ,X ]
When W is type n, the map v tn on W induces a map
v tn : πi (X ; W )→ πi+t|vn|(X ; W ) for i ≥ N.
Definition
The vn-periodic homotopy groups of X with coefficients in W isdefined by
v−1n π∗(X ; W ) = (v t
n)−1π∗(X ; W )
Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View
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The EHP SequencePeriodic Unstable Homotopy Theory
The K(2)-local Goodwillie Tower of SpheresComputation of π∗(ΦK(2)S
3)
Periodic unstable homotopy groupsBousfield-Kuhn functorv1-periodic unstable homotopy groupsBounded torsion phenomenon in chromatic level 2
The Bousfield-Kuhn functor
Let T (n) be the Bousfield class (in the sense of localization) ofv−1n Σ∞W for any type n complex W .
Theorem (Bousfield, Kuhn)
There exists a functor Φn from the category of based spaces tospectrum, such that:
1 If Y is a spectrum, then Φn(Ω∞Y ) ∼= LT (n)Y .
2 For any space X , we have v−1n π∗(X ; W ) = π∗(ΦnX ; W ), for
any type n complex W .
We have the variations ΦK(n) = LK(n)Φn.
Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View
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The EHP SequencePeriodic Unstable Homotopy Theory
The K(2)-local Goodwillie Tower of SpheresComputation of π∗(ΦK(2)S
3)
Periodic unstable homotopy groupsBousfield-Kuhn functorv1-periodic unstable homotopy groupsBounded torsion phenomenon in chromatic level 2
v1-periodic homotopy type of unstable spheres
Let P∞1 = Σ∞BΣp. We can make P∞1 into a CW complex withcells in dimension q − 1, q, 2q − 1, 2q, . . . , where q = 2(p − 1).Define P2k
1 to be the kq-skeleton of P∞1 , which has cells indimension q − 1, q, . . . , kq − 1, kq.
Theorem (Mahowald-Thompson)
ΦK(1)S2k+1 is homotopy equivalent to LK(1)P2k1 .
Remark
At an odd prime, we have LK(1)P∞1 ∼= LK(1)S, and
LK(1)P2k1∼= Σ−1S/pk .
Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View
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The EHP SequencePeriodic Unstable Homotopy Theory
The K(2)-local Goodwillie Tower of SpheresComputation of π∗(ΦK(2)S
3)
Periodic unstable homotopy groupsBousfield-Kuhn functorv1-periodic unstable homotopy groupsBounded torsion phenomenon in chromatic level 2
vn-torsion in unstable homotopy groups
Theorem (W.)
The group π∗(ΦK(2)S3) is annihilated by v 21 for p ≥ 5.
Remark
The map v 21 : Σ2|v1|ΦK(2)S3 → ΦK(2)S3 is non-trivial because it is
not zero on E2-homology.
Theorem (W.)
The group π∗(ΦK(2)S2k+1) has bounded v1-torsion for p ≥ 5.
Conjecture (generalization of Cohen-Moore-Neisendorfer)
The vn-torsion part of π∗(S2k+1) is annihilated by a fixed power(which depends on k) of vn.
Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View
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The EHP SequencePeriodic Unstable Homotopy Theory
The K(2)-local Goodwillie Tower of SpheresComputation of π∗(ΦK(2)S
3)
The Goodwillie tower of the identityThe Goodwillie derivatives of spheresGoodwillie differentials on En-cohomology
Section 3
The K (2)-local Goodwillie Tower of Spheres
Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View
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The EHP SequencePeriodic Unstable Homotopy Theory
The K(2)-local Goodwillie Tower of SpheresComputation of π∗(ΦK(2)S
3)
The Goodwillie tower of the identityThe Goodwillie derivatives of spheresGoodwillie differentials on En-cohomology
The Goodwillie tower
For any based space X , we can construct a tower by applyingGoodwillie calculus to the identity functor:
X → · · · → P4 → P3 → P2 → P1 = Ω∞Σ∞X
Theorem (Goodwillie)
1 When X is connected, we have X ∼= lim←−Pi .
2 The fiber Di of Pi → Pi−1 is an infinite loop space.
Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View
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The EHP SequencePeriodic Unstable Homotopy Theory
The K(2)-local Goodwillie Tower of SpheresComputation of π∗(ΦK(2)S
3)
The Goodwillie tower of the identityThe Goodwillie derivatives of spheresGoodwillie differentials on En-cohomology
The Goodwillie derivatives of spheres
For any sphere Sn, we can construct the Goodwillie tower
· · · → P4(Sn)→ P3(Sn)→ P2(Sn)→ P1(Sn)
The derivatives Di (Sn) are the fibers Pi (Sn)→ Pi−1(Sn).
Theorem (Arone, Dwyer, Mahowald)
Let n be odd.
1 For i not a power of p, Di (Sn) is trivial.
2 Dpk (Sn) ∼= Ω∞Σn−kL(k)n, for L(k)n the Steinberg summand
in (BFkp)nρk , the Thom spectrum of the reduced regular
representation of the additive group Fkp .
3 LT (h)L(k) is trivial when k > h.
Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View
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The EHP SequencePeriodic Unstable Homotopy Theory
The K(2)-local Goodwillie Tower of SpheresComputation of π∗(ΦK(2)S
3)
The Goodwillie tower of the identityThe Goodwillie derivatives of spheresGoodwillie differentials on En-cohomology
BP-cohomology of L(1)
Recall that BP∗(BFp) = BP∗[[ξ]]/[p](ξ). So we have:
BP∗(BFkp) = BP∗[ξ1, . . . , ξk ]/[p](ξ1), . . . , [p](ξk)
L(1)1 can be identified with Σ∞BΣp.
Theorem
BP∗L(1)1 is generated by x , x2, x3, . . . subject to the relationspx + v1x2 + · · · = 0, px2 + v1x3 + · · · = 0 . . . .
The unstable filtration (i.e. BP∗L(1)1 ⊃ BP∗L(1)3 ⊃ · · · ) is thefiltration by powers of x .
Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View
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The EHP SequencePeriodic Unstable Homotopy Theory
The K(2)-local Goodwillie Tower of SpheresComputation of π∗(ΦK(2)S
3)
The Goodwillie tower of the identityThe Goodwillie derivatives of spheresGoodwillie differentials on En-cohomology
BP-cohomology of L(k)
In general, define the Dickson-Mui invariants by the formula
Fi =∏
(a1,...,ai )∈Fip\0
(a1ξ1 +F · · ·+F aiξi )
Theorem
BP∗L(2)1 is generated by F1F2,F21 F2, . . . ,F1F 2
2 ,F21 F 2
2 , . . . , subjectto the relations
pF1F2 = v2F1F 22 + · · ·
. . .
v1F1F2 = v2F p+11 F2 + · · ·
v1F 21 F2 = v2F p+2
1 F2 + v2F1F 22 + · · ·
. . .
Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View
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The EHP SequencePeriodic Unstable Homotopy Theory
The K(2)-local Goodwillie Tower of SpheresComputation of π∗(ΦK(2)S
3)
The Goodwillie tower of the identityThe Goodwillie derivatives of spheresGoodwillie differentials on En-cohomology
BP-homology of L(2)
multiplication by p
ee3,1
ee2,1
ee1,1
ee1,2 e
e2,2
ee1,3
multiplication by v1 (at p = 3)
ee3,3
ee2,3
ee1,3
ee1,4 e
e2,4
ee1,5
ee5,1
ee4,1
ee3,1
ee2,1
ee1,1
ee4,2
ee3,2
ee2,2
ee1,2
@@
@@
@@@
@@I
?
Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View
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The EHP SequencePeriodic Unstable Homotopy Theory
The K(2)-local Goodwillie Tower of SpheresComputation of π∗(ΦK(2)S
3)
The Goodwillie tower of the identityThe Goodwillie derivatives of spheresGoodwillie differentials on En-cohomology
The James-Hopf map
The first attaching map L(0)1 → L(1)1 in the Goodwillie tower ofS1 is the Jame-Hopf map
jh : Ω∞Σ∞S0 → Ω∞Σ∞BΣp
which is the adjoint of the projection map
Σ∞Ω∞Σ∞S1 → Σ∞(S1)∧phΣp
using Snaith splitting
Σ∞Ω∞Σ∞S1 ∼= ∨Σ∞(S1)∧ihΣi
Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View
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The EHP SequencePeriodic Unstable Homotopy Theory
The K(2)-local Goodwillie Tower of SpheresComputation of π∗(ΦK(2)S
3)
The Goodwillie tower of the identityThe Goodwillie derivatives of spheresGoodwillie differentials on En-cohomology
James-Hopf map on En-cohomology
We apply the Bousfield-Kuhn functor to the Jame-Hopf map:
ΦK(n)jh : LK(n)S→ LK(n)Σ∞BΣp
Let En be Morava E -theory. The p-series can be written as[p](ξ) = ξq(ξp−1) for any p-typical formal group law. Define thering R = E ∗n [x ]/q(x). Recall that E ∗n L(1)1 = xR.The finite extension E ∗n → R gives a trace map tr : R → E ∗n .
Theorem (W.)
Up to units, the effect of ΦK(n)jh on En-cohomology is
tr
p: xR → E ∗n
Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View
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The EHP SequencePeriodic Unstable Homotopy Theory
The K(2)-local Goodwillie Tower of SpheresComputation of π∗(ΦK(2)S
3)
Section 4
Computation of π∗(ΦK (2)S3)
Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View
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The EHP SequencePeriodic Unstable Homotopy Theory
The K(2)-local Goodwillie Tower of SpheresComputation of π∗(ΦK(2)S
3)
Goodwillie tower of ΦK (2)S3
We have the following diagram in the K (2)-local category:
L(1)21 → L(2)2
1 ⇒ ΦK(2)Ω4S3
↓ ↓ ↓S → L(1)1 → L(2)1 ⇒ ΦK(2)ΩS1
↓ ↓ ↓ ↓S → L(1)3 → L(2)3 ⇒ ΦK(2)Ω3S3
Let E2∗ = E2∗/p, and R = E2∗[y ]/v1 + v2yp = 0.
Theorem (W.)
After applying E2-homology, we can identify the first row with
E2∗v1−→ yR → E2
∧∗Σ−4ΦK(2)S3
Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View
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The EHP SequencePeriodic Unstable Homotopy Theory
The K(2)-local Goodwillie Tower of SpheresComputation of π∗(ΦK(2)S
3)
π∗ΦK (2)S3 at prime 5
This is half of the E∞-page of the Adams-Novikov spectralsequence computing π∗ΦK(2)S3 for p = 5. The other half is the ζmultiple of it.
qh0yv2
qv1h0yv2
qh0yv
62
qv1h0yv
62
qh1y
4q
h1y4v2
2
qh1y
4v32
qh1y
4v42
qh1y
4v52
qg0yv
32
qv1g0yv
32
qg0y
2v2
qg0y
2v22
qg0y
2v32
qg0y
2v42
qg0y
2q
v1g0y2
qg0y
2v52
qv1g0y
2v52
qg1y
4v−12
qg1y
4q
g1y4v2
qg1y
4v22
qg1y
4v42
qg1y
4v52
qg0h1v
−12
qg0h1
qg0h1v2
qg0h1v
32
qg0h1v
42
qg0h1v
22
qv1g0h1v
22
We have a similar chart for other primes p > 5.
Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View