Unstable Homotopy Theory from the Chromatic Point of...

The EHP SequencePeriodic Unstable Homotopy Theory

The K(2)-local Goodwillie Tower of SpheresComputation of π∗(ΦK(2)S

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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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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)




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Definition of EHP sequenceEHP sequence for p = 3Exponent of unstable homotopy groups

Section 1

The EHP Sequence




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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.




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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.




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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




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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




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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 .




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Periodic unstable homotopy groupsBousfield-Kuhn functorv1-periodic unstable homotopy groupsBounded torsion phenomenon in chromatic level 2

Section 2

Periodic Unstable Homotopy Theory




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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.




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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 )




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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.




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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 .




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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.




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The Goodwillie tower of the identityThe Goodwillie derivatives of spheresGoodwillie differentials on En-cohomology

Section 3

The K (2)-local Goodwillie Tower of Spheres




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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.




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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.




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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 .




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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 + · · ·

. . .




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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

?




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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




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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




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Section 4

Computation of π∗(ΦK (2)S3)




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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




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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.


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