An Introduction to Chromatic Homotopy TheoryPart I : Spectra and Localization

Agnes Beaudry

May 14, 2019

Outline of the Course

(I) Spectra and Localization

(II) Complex Orientations and the Morava K -Theories

(III) The Chromatic Filtration

(IV) Morava E -Theory and the Stabilizer Group

Part I – Spectra and Localization

(1) Cohomology Theories

(2) Spectra

(3) Stable Homotopy Category

(4) Bousfield Localization

(5) What is Chromatic Homotopy Theory?

Cohomology with Coefficients in G

ĄHG˚p´q : CWop

` ÝÑ Ab

1. (Homotopy) If f » g , then ĄHG˚pf q “ ĄHG

˚pgq.

2. (Additivity)

ĄHG˚

˜

ž

iPI

Xi

¸

–ź

iPI

ĄHG˚pXi q

3. (Exactness) For A Ď X a subcomplex, the following sequence is exact:

ĄHG˚pX Aq Ñ ĄHG

˚pX q Ñ ĄHG

˚pAq

4. (Suspension) For each n, there is a natural isomorphism

ĄHGnpX q

–ÝÑ ĄHG

n`1pΣX q.

5. (Dimension) ĄHG˚pS0q “ G in ˚ “ 0.

In fact, it is representable

ĄHGnpX q – rX ,KpG , nqs, KpG , nq

»ÝÑ ΩKpG , n ` 1q.

Eilenberg-Steenrod Axioms

A reduced cohomology theory is a functor

rE˚ : CWop` ÝÑ Ab

which satisfies the following axioms

1. (Homotopy) If f » g then rE˚pf q “ rE˚pgq.

2. (Additivity)

rE˚

˜

ž

iPI

Xi

¸

–ź

iPI

rE˚pXi q

3. (Exactness) For A Ď X a subcomplex, the following sequence is exact:

rE˚pX Aq Ñ rE˚pX q Ñ rE˚pAq

4. (Suspension) For each n, there is a natural isomorphism

rEnpX q–ÝÑ rEn`1pΣX q.

5. (Dimension) rE˚pS0q “ G in ˚ “ 0.

E˚pX q “ rE˚pX`q.

The Brown Representability Theorem

Let E be a cohomology theory. There is a sequence of based spaces En, n ě 0with weak equivalences

ωn : En»ÝÑ ΩEn`1.

such thatrEnpX q – rX ,Ens.

The adjunctionrX ,ΩY s – rΣX ,Y s.

gives the suspension isomorphism

rEnpX q – rX ,Ens–

ωn˝´

// rX ,ΩEn`1s – rΣX ,En`1s – rEn`1pΣX q

The Category of Spectra Sp

Objects. An (Ω-)spectrum E is a sequence of based spaces En, n ě 0 withweak equivalences

ωn : En»ÝÑ ΩEn`1

Morphisms. A map of f : E Ñ F is a sequence of maps fn : En Ñ Fn such thatthe diagram commutes:

Enfn //

ωn

Fn

ωn

ΩEn`1

Ωfn`1 // ΩFn`1

We denote the category of spectra by Sp.

Spectrification

For a sequence E “ tEn : n ě 0u and inclusions ωn : En ãÑ ΩEn`1,

LEn “ limÝÑk

ΩkEn`k , Lωn “ limÝÑk

Ωkωn`k

is a spectrum. This is called spectrification.

Ordinary Cohomology with Coefficients in G

HG˚p´q is represented by

HGn “ KpG , nq ωn : KpG , nq»ÝÑ ΩKpG , n ` 1q.

Complex K -Theory

By Bott Periodicity, ΩU » Z ˆ BU and ΩpZ ˆ BUq » U. Complex K -theoryK˚p´q, is represented by

K “ tZˆ BU,U,Zˆ BU,U,Zˆ BU,U, . . .u.

Suspension Spectra

For a based space X , the suspension spectrum is the spectrification of

pΣ8X qn “ ΣnX ωn : ΣnX Ñ ΩΣn`1X

where ωn is the adjoint to the identity

σn : ΣΣnX“ÝÑ Σn`1X .

We often write X for Σ8X . The sphere spectrum is S0 “ Σ8S0.

Complex Cobordism

MU is the spectrification of the sequence

tMUp1q,ΣMUp1q,MUp2q,ΣMUp2q, . . .u

where MUpnq is the Thom space of the canonical complex vector bundle

γn Ñ GrnpC8q » BUpnq.

Note:i˚γn`1 – γn ‘ C

// γn`1

BUpnq

i // BUpn ` 1q

The map ω2n`1 : ΣMUpnq Ñ MUp2nq is adjoint to:

Σ2MUpnq – Thpi˚γn`1qσ2n`1 // Thpγn`1q “ MUpn ` 1q

The map ω2n : MUpnq Ñ ΩΣMUpnq is adjoint to the identity.

Homotopy Groups

If E is a spectrum, then the rth homotopy group of E is

πrE “ limÝÑn

πr`nEn.

A map f : E Ñ F is a weak equivalence if π˚f is an isomorphism.

Examples

§ The stable homotopy groups of spheres are are

πsr “ πrΣ8S0 – lim

ÝÑn

πr`nSn.

§ π˚HA – A concentrated in ˚ “ 0.

§ π˚K – Zrβ˘1s for β P π2K “ KpCP1q the Bott class, i.e.,

π2rK “ Ztβr u, π2r`1K “ 0.

§ π˚MU – Zrx1, x2, . . .s for xn P π2nMU related to rCPns.

π0MU “ Zt1u, π1MU “ 0, π2MU “ Ztx1u, π4MU “ Ztx21 , x2u.

Homotopy Category

Let C be a category and W a subcategory such that

‚ All isomorphisms of C are in W,

‚ If 2 out of 3 of tf , g , g ˝ f u are in W, then so is the third.

The homotopy category of C is a category HopC,Wq and a functor

ι : C Ñ HopC,Wq

such that, for F : C Ñ D which maps W to isomorphisms, there is

C

ι

F // D

HopC,WqD!FW

;;

such that F–ÝÑ FW ˝ ι.

Stable Homotopy Category

SH “ HopSp,WS0 q for WS0 the weak equivalences is called the stable homo-topy category.

§ rX ,Y s :“ SHpX ,Y q are abelian groups

§ Finite products and coproducts are equivalent in SH§ Closed symmetric monoidal with unit S0:

´^´ : SHˆ SHÑ SH F p´,´q : SHop ˆ SHÑ SH

F pX ^Y ,Zq – F pX ,F pY ,Zqq

§ Triangulated: Σ: SHÑ SH with

ΣX “ X ^S1 Σ´1X “ F pS1,X q.

§ Distinguished triangles

Xf // Y // Cf

// ΣX

where Cf “ Y Yf pX ^r0, 1sq is the cofiber of f .

§ Long exact sequences for r´,Z s and rZ ,´s. For example:

. . . // πnX // πnY // πnCf// πn´1X // . . .

Models for Spectra

There are other choices pC,Wq with SH “ HopC,Wq. In particular, there areclosed symmetric monoidal models for Sp.

Homology and Cohomology

If E is a spectrum, then E -homology is the functor rE˚ : SHÑ Ab

X ÞÑ rEnpX q :“ πnpE ^X q

and E -cohomology is the functor E˚ : SHop Ñ Ab

X ÞÑ rEnpX q :“ π´nF pX ,Eq “ rX ,ΣnE s.

We letEn :“ rEnpS

0q “ πnE “ rE´npS0q “: E´n

Stable Homotopy Groups as a Homology Theory

For E “ S0, this gives:ĂS0˚pX q “ π˚pX q

Weak equivalences are ĂS0˚-isomorphisms and SH is obtained from Sp by

inverting these.

Bousfield Localization

A map f : X Ñ Y is an E -equivalences if

rE˚pf q : rE˚pX q–ÝÑ rE˚pY q

Let WE Ď Sp be the subcategory of E -equivalences and

SHE :“ HopSp,WE q.

§ X is E -acyclic if rE˚pX q “ 0.

§ Y is E -local if rX ,Y s “ 0 when X is E -acyclic.

Let SpE be the full subcategory whose objects are E -local spectra. In fact:

SHE – HopSpE ,W X SpE q.

Exercise

A map f : X Ñ Y in SpE is a weak equivalence iff it is an E -equivalence.

Universal Property of Localization

An E -localization is an E -equivalence η : X Ñ LEX for LEX P SpE . Theseexist and are unique in SH:

LE : SHÑ SHE Ď SH, η : 1SH Ñ LE .

If f : X Ñ Y with Y P SpE , then

Xf //

η

Y

LEX

fE

==

and fE is unique in SH. There is a distinguished triangle

CEX Ñ X Ñ LEX

where CEX is the terminal E˚-acyclic spectrum with a map to X .

Exercise

There is a natural transformation LF_E Ñ LE and,

LE » LELE_F » LE_FLE .

Localization and Completion

For G an abelian group, let SG a a Moore spectrum

š

jPJ S0trju // š

iPI S0tgiu // SG // Σ

š

jPJ S0trju

for a presentationÀ

jPJ Ztrju ÑÀ

iPI Ztgiu Ñ G .

§ If G “ Zppq or Q, then LSGX » SG ^X and

π˚LSGX – π˚X b G .

The p-localization of X is:

Xppq :“ LSZppqX

The rationalization of X is:

XQ :“ LSQX

§ If the groups π˚X are finitely generated, then

π˚LSZpX – π˚X b Zp

where Zp is the p-adic integers. The p-completion of X is

Xp :“ LSZpX .

The Sphere

For the p-local sphere Sppq “ LSZppqS ,

πnSppq –

#

Zppq n “ 0

Torppπsnq n ą 0.

For the rarional sphere SQ “ LSQS ,

πnSQ –

#

Q n “ 0

0 n ą 0.

Connective Spectra

A spectrum is connective if πrX “ 0 for r ă 0. For such X

LHGX » LSGX .

Adams Operations

For k ě 0, there are unstable operations

ψk : K˚pX q Ñ K˚pX q.

They give stable operations

ψk : Kp Ñ Kp

for k P Zˆp The action of ψk on π˚Kp “ Zprβ˘1s is determined by

ψk pβq “ kβ.

K -Theory Localization

Let p be odd and KZp “ K ^SZp so that

π˚KZp – Zprv˘11 s.

There is a distinguished triangle

LKZpS0 // Kp

ψ`´1 // Kp // ΣLKZpS0

for ` a topological generator of Zˆp .

Image of J

Real Bott Periodicity gives π3`4kO – Z. The J homomorphism

J : πnO Ñ πsn

has image

n 3 7 11 15 19 23 27 31 35 39 43Z24 Z240 Z504 Z480 Z264 Z65520 Z24 Z16320 Z28728 Z13200 Z552

Lower Bounds on Im J Section 4.1 99

4.1 Lower Bounds for Im J

After starting this section with the definition of the J–homomorphism, we will

use a homomorphism K(X)→H∗(X;Q) known as the Chern character to show that

an/2 is a lower bound on the order of the image of J in dimension 4n − 1. Then

using real K–theory this bound will be improved to an , and we will take care of the

cases in which the domain of the J–homomorphism is Z2 .

The simplest definition of the J–homomorphism goes as follows. An element

[f ] ∈ πi(O(n)) is represented by a family of isometries fx ∈ O(n) , x ∈ Si , with fx

the identity when x is the basepoint of Si . Writing Sn+i as ∂(Di+1×Dn) = Si×Dn ∪

Di+1×Sn−1 and Sn as Dn/∂Dn , let Jf(x,y) = fx(y) for (x,y) ∈ Si×Dn and let

Jf(Di+1×Sn−1) = ∂Dn , the basepoint of Dn/∂Dn . Clearly f ≃ g implies Jf ≃ Jg ,

so we have a map J :πi(O(n))→πn+i(Sn) . We will tacitly exclude the trivial case

i = 0.

Proposition 4.1. J is a homomorphism.

Proof: We can view Jf as a map In+i→Sn = Dn/∂Dn which on Si×Dn ⊂ In+1 is

given by (x,v)!fx(v) and which sends the complement of Si×Dn to the basepoint

∂Dn . Taking a similar view of Jg , the sum Jf + Jg is obtained by juxtaposing these

two maps on either side of a hyperplane. We may assume fx is the identity for x in

the right half of Si and gx is the identity for x in the left half of Si . Then we obtain

a homotopy from Jf + Jg to J(f + g) by moving the two Si×Dn ’s together until

they coincide, as shown in the figure below. )⊓

→ →We know that πi(O(n)) and πn+i(S

n) are independent of n for n > i + 1, so

we would expect the J–homomorphism defined above

Exercise

Compute π˚LKZpS0 and compare with the p-component of impJq.

Next Time

Fix a prime p. There are spectra Kp0q,Kp1q,Kp2q, . . . ,Kp8q called theMorava K -theories:

LnX :“ LKp0q_..._KpnqX

In fact,LKp0q – LHQ LKp1q – LKZp LKp8q – LHZp

The chromatic convergence theorem (Hopkins–Ravenel) states:

S0ppq – lim

ÐÝn

LnS0

in SH. The nth chromatic layer is LnS0.

Chromatic homotopy theory studies the S0ppq

via this filtration.

π˚Sp2q (Illustration by Isaksen)

3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 350

1

2

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The E∞-page of the classical Adams spectral sequence

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70

0

2

4

6

8

10

12

14

16

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36

h0 h1 h2 h3

c0

Ph1 Ph2

h23

d0 h30h4

h1h4

Pc0

P 2h1

h2h4

c1

P 2h2

g

Pd0

h4c0

h20i

P 2c0

P 3h1 P 3h2

d20

h24

n

h100 h5

h1h5

d1

q

P 3c0

p

P 4h1

h0h2h5

e20

P 4h2

t

h22h5

x

h20h3h5

h1h3h5

h5c0

h3d1

u

P 2h20i

f1

Ph1h5

g2

P 4c0

z

P 5h1

Ph2h5

d30

P 5h2

g2

h23h5

h5d0

w

B1

N

d0l

Ph5c0

e0r

Pu

h70Q′

B2

d0e20

P 5c0

P 6h1

h5c1

C

h3g2

gn

P 6h2

d1g

e0m

x′

d0u

h0h5i

e20g

P 4h20i

P 6c0

P 7h1

h1Q2

B21

d0w

P 7h2

B3

g3

d20l

D3

A′

d0e0r

h25

h5n

E1 + C0

Rh1X1

d20e2

0

h1H1

X2C′

h250 h6

h1h6

h3Q2

q1

P 7c0

k1

B23

Ph5j

gw

P 8h1

r1

B5 + D′2

d0e0m

Q3

X3

C11

d0x′ e0gr

d20u

P 8h2

d2

h22H1

h3A′

h30G21

h0h4x′

h22h6

p′

D′3

h3(E1 + C0)

h3h6

p1

h2Q3

h1h3H1

d1e1

h1W1

π˚Sp2q (Illustration by Isaksen)

π˚Sp2q (Illustration by Isaksen)

Telescope Conjecture (Ravenel)

The first n-rays are detected by LnS0.

Chromatic Splitting Conjecture (Hopkins)

The gluing data for the chromatic layers is simple.

Thank you!