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