The Kahn-Priddy Theoremjkjaer/Presentation2.pdf · 1937 Freudenthals Suspension Theorem begins...

department of mathemat i cal sc i ence s

university of copenhagen

The Kahn-Priddy Theorem

Jens Jakob Kjær

Presentation of Master Thesis

February 7, 2014

un iver s i ty of copenhagen department of mathemat i cal sc i ence s

Contents

1 History and Overview

2 Stable Homotopy Theory

3 Constructions of P−k

4 The Kahn-Priddy TheoremA LemmaThe Proof

5 The Spectral Sequence

/20 Jens Jakob Kjær The Kahn-Priddy Theorem February 7, 2014

History and OverviewHistory

• 1937 Freudenthals Suspension Theorem begins stablehomotopy theory

• 1972 Kahn and Priddy announce theorem linking stablehomotopy groups of innite real projective space to thestable homotopy groups of spheres [KP72]

• 1985 Jones strengthens the result [Jon85]

• 1990 Miller gives postcard length proof of Jones'strengthening [Mil90]

History and OverviewOverview

Twofold goal of the thesis

1 give an in-depth proof of Jones' theorem based onMiller's paper, and

2 give a detailed presentation (with some illustratingcalculations) of the part of the spectral sequence thatthis illuminates.

Theorem (The Kahn-Priddy Theorem)

There exists a morphism τ ′ : RP∞ → S0 which is surjective

on positive 2-localized stable homotopy groups.

Stable Homotopy Theory

Denition

A spectrum X is a sequence of pointed topological spaces Xn

with pointed continuous maps σn : ΣXn → Xn+1

Denition

A morphism of spectra f : X → Y is a sequence of

continuous pointed maps fn : Xn → colimkΩYn+k such that

Ωfn+1 = fn.

Denition

X a pointed space then Σ∞X the suspension spectrum

(Σ∞X)n = ΣnX.

Denition

Ωfn+1 = fn.

Denition

(Σ∞X)n = ΣnX.

Denition

Ωfn+1 = fn.

Denition

(Σ∞X)n = ΣnX.

Denition

Ωfn+1 = fn.

Denition

(Σ∞X)n = ΣnX.

Denition

Sn := ΣnΣ∞S0.

Denition

Sn := ΣnΣ∞S0.

Denition

Sn := ΣnΣ∞S0.

Denition

Let Ho(Spec) be the category with objects spectra

and HomHo(Spec)(X,Y ) = [X,Y ]

Proposition

πtΣ∞X ∼= colimkπ

unstt+k ΣkX

Denition

Proposition

unstt+k ΣkX

Denition

Proposition

unstt+k ΣkX

Denition

Proposition

unstt+k ΣkX

Stable Homotopy TheoryS-duality

Denition

We say that X is S-dual to X∗ if there is morphism

µ : X ∧X∗ → S0 such that DU,V : [U, V ∧X]→ [U ∧X∗, V ]and DU,V [U,X∗ ∧ V ]→ [X ∧ U, V ] are isomorphisms for all

spectra U, V .

If X,X∗, Y, Y ∗ are S-dual and f : X → Y then there is adual map in [Y ∗, X∗].

Proposition

The S-dual of St is S−t. Given α : St → Ss then the dual is

α : S−s → S−t.

Denition

spectra U, V .

Proposition

α : S−s → S−t.

Denition

spectra U, V .

Proposition

α : S−s → S−t.

Denition

spectra U, V .

Proposition

α : S−s → S−t.

Constructions of P−k

Denition

Pn+kn = RPn+k/RPn−1

If k ∈ N φ(k) is the number of integers 0 < i ≤ k such thati ≡ 0, 1, 2, 4 (mod 8)

Denition

For n ∈ Z and k ∈ N0 we dene the spectra

Pn+kn = Σn−rΣ∞P r+kr for any r ≡ n modulo 2φ(k), r ≥ 0.

Denition

Example

• P k1 = Σ∞RP k

• P k0 = Σ∞RP k+• P−1−3 = Σ−4Σ∞P 3

Proposition

The S-dual of P k−1−n is ΣPn−1−k

Example

• P k1

= Σ∞RP k

• P k0 = Σ∞RP k+• P−1−3 = Σ−4Σ∞P 3

Proposition

Example

• P k0 = Σ∞RP k+• P−1−3 = Σ−4Σ∞P 3

Proposition

Example

• P k0

= Σ∞RP k+• P−1−3 = Σ−4Σ∞P 3

Proposition

Example

• P k0 = Σ∞RP k+

• P−1−3 = Σ−4Σ∞P 31

Proposition

Example

• P k0 = Σ∞RP k+• P−1−3

= Σ−4Σ∞P 31

Proposition

Example

• P k0 = Σ∞RP k+• P−1−3 = Σ−4Σ∞P 3

Proposition

Example

• P k0 = Σ∞RP k+• P−1−3 = Σ−4Σ∞P 3

Proposition

Denition

Given a spectra X we can dene Dk2X to be the spectra with

(Dk2X)n = Sk×(X∧X)n

Z/2 /RP k × ∗ and if

σ : Σ(X ∧X)n → (X ∧X)n+1 is the structure map of

X ∧X, then idSk × σ induces the structure maps of Dk2X.

Proposition

Pn+kn = Σ−nDk(Sn)

There is a natural morphism ϕZ : Z ∧Dk2(X)→ Dk

2(Z ∧X)

Denition

Proposition

2(Z ∧X)

Denition

Proposition

2(Z ∧X)

Denition

Proposition

2(Z ∧X)

Constructions of P−kMorphisms of interest

Denition

• P kn → P k+1n

• c : P kn → P kn+1

• π : P k−10 → S0

By taking homotopy colimit over the inclusions we get PnBy S-duality we get morphism ι : S−1 → P−1−k → P−k

Cober sequence S−1ι→ P−1 → P0

τ→ S0

Cober sequences S−k → P−kc→ P−k+1

The dual of P k−1−n → P k−n is c : Pn−1−k → Pn−1−k+1

Denition

• P kn → P k+1n

• c : P kn → P kn+1

• π : P k−10 → S0

τ→ S0

Denition

• P kn → P k+1n

• c : P kn → P kn+1

• π : P k−10 → S0

τ→ S0

Denition

• P kn → P k+1n

• c : P kn → P kn+1

• π : P k−10 → S0

τ→ S0

Denition

• P kn → P k+1n

• c : P kn → P kn+1

• π : P k−10 → S0

By taking homotopy colimit over the inclusions we get Pn

By S-duality we get morphism ι : S−1 → P−1−k → P−k

τ→ S0

Denition

• P kn → P k+1n

• c : P kn → P kn+1

• π : P k−10 → S0

τ→ S0

Denition

• P kn → P k+1n

• c : P kn → P kn+1

• π : P k−10 → S0

τ→ S0

Denition

• P kn → P k+1n

• c : P kn → P kn+1

• π : P k−10 → S0

τ→ S0

Denition

• P kn → P k+1n

• c : P kn → P kn+1

• π : P k−10 → S0

τ→ S0

Denition

• P kn → P k+1n

• c : P kn → P kn+1

• π : P k−10 → S0

τ→ S0

There exists a morphism τ ′ : P1 → S0 which is surjective on

positive 2-localized homotopy groups

Theorem (Jones' Kahn-Priddy Theorem)

If s < t then for any α ∈ πt−1S−1 then

St−1α→ S−1

ι→ P−s−1 is trivial.

St−1α→ S−1

The Kahn-Priddy TheoremA lemma

Denition

Given α ∈ πtS0 then Q(α) : Pt → S−t it the t'thdesuspension of

D2(St)D2(α)→ D2(S

0) = P0π→ S0

Take α ∈ πtS0, and π the pinch map, then the following

diagram commutes for 0 ≤ t < r

πc(r) // S0

PtQ(α) // S−t

Denition

0) = P0π→ S0

πc(r) // S0

PtQ(α) // S−t

Denition

0) = P0π→ S0

πc(r) // S0

PtQ(α) // S−t

The Kahn-Priddy TheoremProof of Lemma

St+r ∧D2(S−r)

α∧1

ϕSr //

ϕSt+r

St ∧D2(S0)π //

α∧1

Sr ∧D2(S−r)

((D2(S

%%D2(S

t)Q(α)

//D2(α)

The Kahn-Priddy TheoremProof of Lemma

St+r ∧D2(S−r)

α∧1

ϕSr //

ϕSt+r

St ∧D2(S0)π //

α∧1

Sr ∧D2(S−r)

((D2(S

%%D2(S

t)Q(α)

//D2(α)

The Kahn-Priddy TheoremProof of Theorem

St−1α→ S−1

πc(r) // S0

PtQ(α) // S−t

The Kahn-Priddy TheoremProof of Theorem

St−1α→ S−1

πc(r) // S0

PtQ(α) // S−t

The Spectral Sequence

Theorem

There is a spectral sequence with

Es,t1 := πt−s+1S−s+1 = πtS

0 and dr : Es,tr → Es+r,t+r−1

converging to π∗S−12

This is the spectral sequence derived from the tower

P−2 //

P−1 //

Theorem

P−2 //

P−1 //

Theorem

P−2 //

P−1 //

The Spectral SequenceThe E1-page

−s−5 −4 −3 −2 −1 0 1

ZZZZZZZ

Z/2Z/2Z/2Z/2Z/2Z/2Z/2

0000000

The Spectral SequenceFiltration

Denition

Dene M sπt ⊂ πtS02by

M sπt = Ker [ι∗ : πt−1S−12→ πt−1P−s]

M s−1πt−s+1/Msπt−s+1

∼= Es,t∞

Corollary

M sπt = πt−1S−12

whenever s ≤ t.

Denition

∼= Es,t∞

Corollary

whenever s ≤ t.

Denition

∼= Es,t∞

Corollary

whenever s ≤ t.

Denition

∼= Es,t∞

Corollary

whenever s ≤ t.

The Spectral SequenceThe E∞-page

−s−5 −4 −3 −2 −1 0 1

XXXXXX

Bibliography

John D. S. Jones, Root Invariants, Cup-r-Products andThe Kahn-Priddy Theorem, Bulletin of the LondonMathematical Society 17 (1985), 479483.

Daniel S Kahn and Stewart B Priddy, Applications of thetransfer to stable homotopy theory, Bull. Amer. Math.Soc 78 (1972), no. 1972, 135146.

Haynes Miller, On Jones's Kahn-Priddy Theorem,Homotopy Theory and Related Topics - Proceedings ofthe International Conference held at Kinosaki, Japan,August 1924, 1988 (Mamoru Mimura, ed.),Springer-Verlag, 1990, pp. 210218.

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