Upload
others
View
12
Download
0
Embed Size (px)
Citation preview
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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