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J-homomorphism and chromatic homotopy

Contents

Context

Stable Homotopy theory

Higher algebra

This entry explains the J-homomorphism, states how its image is the first (chromatic) layer of the sphere spectrum; and then motivated by this explains some basic notions of chromatic homotopy theory, notably the origin of the general EE-Adams spectral sequence.

Contents

I) The J-homomorphism

The J-homomorphism is a map from the homotopy groups of the stable orthogonal group (which are completely understood) to the stable homotopy groups of spheres (which in their totality are hard to compute).

On groups

Definition

For nn \in \mathbb{N} regard the nn-sphere (as a topological space) as the one-point compactification of the Cartesian space n\mathbb{R}^n

S n( n) *. S^n \simeq (\mathbb{R}^n)^\ast \,.
Remark

Since the process of one-point compactification is a functor on proper maps, hence on homeomorphisms, via def. the nn-sphere inherits from the canonical action of the orthogonal group O(n)O(n) on n\mathbb{R}^n an action

O(n)×S nS n O(n) \times S^n \longrightarrow S^n

(by continuous maps) which preserves the base point (the “point at infinity”).

For definiteness we distinguish in the following notationally between

  1. the nn-sphere S nTopS^n \in Top regarded as a topological space;

  2. its homotopy type Π(S n)L wheTop\Pi(S^n) \in L_{whe} Top \simeq ∞Grpd given by its fundamental ∞-groupoid.

Similarly we write Π(O(n))\Pi(O(n)) for the homotopy type of the orthogonal group, regarded as a group object in an (∞,1)-category in ∞Grpd (using that the shape modality Π\Pi preserves finite products).

Definition

For nn \in \mathbb{N} write H(n)H(n) for the automorphism ∞-group of homotopy self-equivalences S nS nS^n \longrightarrow S^n, hence

H(n)Aut Grpd */(Π(S n)). H(n) \coloneqq Aut_{\infty Grpd^{\ast/}}(\Pi(S^n)) \,.
Remark

The ∞-group H(n)H(n), def. , constitutes the two connected components of the nn-fold based loop space Ω nS n\Omega^n S^n corresponding to the homotopy groups ±1π n(S n)\pm 1 \in \pi_n(S^n).

Definition

Via the presentation of ∞Grpd by the cartesian closed model structure on compactly generated topological spaces (and using that S nS^n and O(n)O(n) and hence their product are compact) we have that for nn \in \mathbb{N} the continuous action of O(n)O(n) on S nS^n of remark , which by cartesian closure is equivalently a homomorphism of topological groups of the form

O(n)Aut Top */(S n), O(n) \longrightarrow Aut_{Top^{\ast/}}(S^n) \,,

induces a homomorphism of ∞-groups of the form

Π(O(n))Aut Grpd */(Π(S n)). \Pi(O(n)) \longrightarrow Aut_{\infty Grpd^{\ast/}}(\Pi(S^n)) \,.

This in turn induces for each ii \in \mathbb{N} homomorphisms of homotopy groups of the form

π i(O(n))π i(Ω nS n)π n+i(S n). \pi_i(O(n)) \longrightarrow \pi_i(\Omega^n S^n) \simeq \pi_{n+i}(S^n) \,.
Remark

By construction, the homomorphisms of remark are compatible with suspension in that for all nn \in \mathbb{N} the diagrams

O(n) Aut Top */(S n) O(n+1) Aut Top */(S n+1) \array{ O(n) &\longrightarrow& Aut_{Top^{\ast/}}(S^n) \\ \downarrow && \downarrow \\ O(n+1) &\longrightarrow& Aut_{Top^{\ast/}}(S^{n+1}) }

in Grp(Top)Grp(Top) commute, and hence so do the diagrams

Π(O(n)) Aut Grpd */(Π(S n)) Π(O(n+1)) Aut Grpd */(Π(S n+1)) \array{ \Pi(O(n)) &\longrightarrow& Aut_{\infty Grpd^{\ast/}}(\Pi(S^n)) \\ \downarrow && \downarrow \\ \Pi(O(n+1)) &\longrightarrow& Aut_{\infty Grpd^{\ast/}}(\Pi(S^{n+1})) }

in Grp(Grpd)Grp(\infty Grpd), up to homotopy.

Therefore one can take the direct limit over nn:

Definition

By remark there is induced a homomorphism

J i:π (O)π (𝕊) J_i \;\colon\; \pi_\bullet(O) \longrightarrow \pi_\bullet(\mathbb{S})

from the homotopy groups of the stable orthogonal group to the stable homotopy groups of spheres. This is called the J-homomorphism.

On classifying spaces

Remark

Since the maps of def. are ∞-group homomorphisms, there exists their delooping

BJ:BOBGL 1(𝕊)=BH. B J \;\colon\; B O \longrightarrow B GL_1(\mathbb{S}) = B H \,.
Remark

Here GL 1(𝕊)GL_1(\mathbb{S}) is the ∞-group of units of the sphere spectrum.

This map BJB J is the universal characteristic class of stable vector bundles with values in spherical fibrations:

Definition

For VXV \to X a vector bundle, write S VS^V for its fiber-wise one-point compactification. This is a sphere bundle, a spherical fibration. Write 𝕊 V\mathbb{S}^V for the XX-parameterized spectrum which is fiberwise the suspension spectrum of S VS^V.

It is immediate that:

Proposition

For VXV \to X a vector bundle classified by a map XBOX \to B O, the corresponding spherical fibration 𝕊 V\mathbb{S}^V, def. , is classified by XBOBJBGL 1(𝕊)X \to B O \stackrel{B J}{\longrightarrow} B GL_1(\mathbb{S}), def. .

II) The image of the J-homomorphism

Since the J-homomorphism maps from something well-understood to something hard to understand, it is of interst to characterize its image, “the image of J”.

Explicit description

The following characterization of the image of the J-homomorphism on homotopy groups derives from a statement that was first conjectured in (Adams 66) – and since called the Adams conjecture – and then proven in (Quillen 71, Sullivan 74).

Remark

By the discussion at orthogonal group – homotopy groups we have that the homotopy groups of the stable orthogonal group are

nmod8n\;mod\; 801234567
π n(O)\pi_n(O) 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z}

Because all groups appearing here and in the following are cyclic groups, we instead write down the order

nmod8n\;mod\; 801234567
|π n(O)|{\vert\pi_n(O)\vert}221\infty111\infty

For the following statement it is convenient to restrict to J-homomorphism to the stable special orthogonal group SOS O, which removes the lowest degree homotopy group in the above

nmod8n\;mod\; 801234567
π n(SO)\pi_n(S O)0 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z}
nmod8n\;mod\; 801234567
|π n(SO)|{\vert\pi_n(S O)\vert}121\infty111\infty
Theorem

The stable homotopy groups of spheres π n(𝕊)\pi_n(\mathbb{S}) are the direct sum of the (cyclic) image im(J| SO)im(J|_{SO}) of the J-homomorphism, def. , applied to the special orthogonal group and the kernel of the Adams e-invariant.

Moreover,

  • for n=0modn = 0 \;mod \; and n=1mod8n = 1 \;mod \; 8 and nn positive the J-homomorphism π n(J):π n(SO)π n(𝕊)\pi_n(J) \colon \pi_n(S O) \to \pi_n(\mathbb{S}) is injective, hence its image is 2\mathbb{Z}_2,

  • for n=3mod8n = 3\; mod\; 8 and n=7mod8n = 7 \; mod \; 8 hence for n=4k1n = 4 k -1, the order of the image is equal to the denominator of B 2k/4kB_{2k}/4k in its reduced form, where B 2kB_{2k} is the Bernoulli number

  • for all other cases the image is necessarily zero.

The characterization of this image is due to (Adams 66, Quillen 71, Sullivan 74). Specifically the identification of J(π 4n1(SO))J(\pi_{4n-1}(S O)) is (Adams 65a, theorem 3.7 and the direct summand property is (Adams 66, theorems 1.1-1.6.). That the image is a direct summand of the codomain is proven for instance in (Switzer 75, end of chapter 19).

A modern version of the proof, using methods from chromatic homotopy theory, is surveyed in some detail in (Lorman 13).

The statement of the theorem is recalled for instance as (Ravenel, chapter 1, theorem 1.1.13). Another computation of the image of JJ is in (Ravenel, chapter 5, section 3).

Remark

The order of J(π 4k1O)J(\pi_{4k-1} O) in theorem is for low kk given by the following table

k12345678910
|J(π 4k1(O))|\vert J(\pi_{4k-1}(O))\vert2424050448026465,5202416,32028,72813,200

See for instance (Ravenel, Chapt. 1, p. 5).

Remark

Therefore we have in low degree the following situation

nn012345678910111213141516
Whitehead tower of orthogonal grouporientationspinstringfivebraneninebrane
homotopy groups of stable orthogonal groupπ n(O)\pi_n(O) 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z} 2\mathbb{Z}_2
stable homotopy groups of spheresπ n(𝕊)\pi_n(\mathbb{S})\mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_2 24\mathbb{Z}_{24}00 2\mathbb{Z}_2 240\mathbb{Z}_{240} 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 2 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2 6\mathbb{Z}_6 504\mathbb{Z}_{504}0 3\mathbb{Z}_3 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 480 2\mathbb{Z}_{480} \oplus \mathbb{Z}_2 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2
image of J-homomorphismim(π n(J))im(\pi_n(J))0 2\mathbb{Z}_20 24\mathbb{Z}_{24}000 240\mathbb{Z}_{240} 2\mathbb{Z}_2 2\mathbb{Z}_20 504\mathbb{Z}_{504}000 480\mathbb{Z}_{480} 2\mathbb{Z}_2
Example

The following tables show the p-primary components of the stable homotopy groups of spheres for low values, the image of J appears as the bottom row.

Here the horizontal index is the degree nn of the stable homotopy group π n\pi_n. The appearance of a string of kk connected dots vertically above index nn means that there is a direct summand primary group of order p kp^k. See example below for illustration. (These tables are taken from (Hatcher), where in turn they were generated based on (Ravenel 86)).

p=2p = 2-primary component (e.g. Ravenel 86, theorem 3.2.11, figure 4.4.46)

stable homotopy groups of spheres at 2

p=3p = 3-primary component

stable homotopy groups of spheres at 3

p=5p = 5-primary component

stable homotopy groups of spheres at 5

We illustrate how to read these tables:

Example

The finite abelian group π 3(𝕊) 24\pi_3(\mathbb{S}) \simeq \mathbb{Z}_{24} decomposes into primary groups as 8 3\simeq \mathbb{Z}_8 \oplus \mathbb{Z}_3. Here 8=2 38 = 2^3 corresponds to the three dots above n=3n = 3 in the first table, and 3=3 13 = 3^1 to the single dot over n=3n = 3 in the second.

The finite abelian group π 7(𝕊) 240\pi_7(\mathbb{S}) \simeq \mathbb{Z}_{240} decomposes into primary groups as 16 3 5\simeq \mathbb{Z}_{16} \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_5. Here 16=2 416 = 2^4 corresponds to the four dots above n=7n = 7 in the first table, and 3=3 13 = 3^1 to the single dot over n=7n = 7 in the second and 5=5 15 = 5^1 to the single dot over n=7n = 7 in the third table.

The finite abelian group π 11(𝕊) 504\pi_11(\mathbb{S}) \simeq \mathbb{Z}_{504} has primary group-decomposition 2 3 3 2 7\cdots \simeq \mathbb{Z}_{2^3} \oplus \mathbb{Z}_{3^2} \oplus \mathbb{Z}_7 and so this corresponds to the three connected dots over n=11n = 11 in the first table and the two connected dots over n=11n = 11 in the second (and there will be one dot over n=11n = 11 in the fourth table for p=7p = 7 not shown here).

The groups π 1(𝕊)π 2(𝕊)π 6(𝕊)π 10(𝕊) 2\pi_1(\mathbb{S}) \simeq \pi_2(\mathbb{S}) \simeq \pi_6(\mathbb{S}) \simeq \pi_{10}(\mathbb{S}) \simeq \mathbb{Z}_2 correspond to the single dots over n=1,2,6,10n = 1,2,6,10 in the first table, respectively.

The group π 8(𝕊) 2 2\pi_8(\mathbb{S}) \simeq \mathbb{Z}_2 \oplus \mathbb{Z}_2 corresponds to the two unconnected dots over n=8n = 8 in the first table.

Similarly the group π 9(𝕊) 2 2 2\pi_9(\mathbb{S}) \simeq \mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2 corresponds to the three unconnected dots above n=9n = 9 in the first table.

Chromatic formulation

The above tables, example , suggest that the image of the J-homomorphism is in some sense the “lowest order layer” of the stable homotopy groups of spheres. This is made precise by the following characterization of the image in stable homotopy theory. We bluntly state this here and give all the relevant definitions below.

\,

Write E(1)E(1) for the first Morava E-theory spectrum at given prime number pp. Write L E(1)𝕊L_{E(1)}\mathbb{S} for the Bousfield localization of spectra of the sphere spectrum at E(1)E(1).

Theorem

The homotopy groups of the E(1)E(1)-localized sphere spectrum are

π nL E(1)𝕊{ ifn=0 p/ p ifn=2 p k+1 ifn+1=(p1)p kmwithm0modp 0 otherwise. \pi_n L_{E(1)} \mathbb{S} \simeq \left\{ \array{ \mathbb{Z} & if\; n = 0 \\ \mathbb{Q}_p/\mathbb{Z}_p & if\; n= -2 \\ \mathbb{Z}_{p^{k+1}} & if\; n+1 = (p-1)p^k m \;with\; m \neq 0\;mod\;p \\ 0 & otherwise } \right. \,.

This appears as (Lurie 10, theorem 6)

Definition

Write 𝕊 p\mathbb{S}_p for the p-localization of the sphere spectrum. For nn \in \mathbb{Z}, write im(J) nim(J)_n for the image of the pp-localized J-homomorphism

J:π n(O)π n(𝕊)π n(𝕊 (p)). J \;\colon\; \pi_n(O) \longrightarrow \pi_n(\mathbb{S}) \longrightarrow \pi_n(\mathbb{S}_{(p)}) \,.
Theorem

For nn \in \mathbb{N}, the further Bousfield localization at Morava E(1)-theory 𝕊 (p)L E(1)𝕊\mathbb{S}_{(p)} \longrightarrow L_{E(1)}\mathbb{S} induces a isomorphism

im(J) nπ n(L E(1)𝕊) im(J)_n \stackrel{\simeq}{\longrightarrow} \pi_n (L_{E(1)} \mathbb{S})

between the image of the JJ-homomorphism and the E(1)E(1)-local stable homotopy groups of spheres.

In this form this appears as (Lurie 10, theorem 7). See also (Behrens 13, section 1).

Corollary

The E(1)E(1)-localization map is surjective on non-negative homotopy groups:

π n(𝕊 (p))π n(L E(1)𝕊). \pi_n(\mathbb{S}_{(p)}) \longrightarrow \pi_n(L_{E(1)} \mathbb{S}) \,.

For review see also (Lorman 13). That JJ factors through L K(1)𝕊L_{K(1)}\mathbb{S} is in (Lorman 13, p. 4)

Remark

Hence: the image of JJ is essentially the first chromatic layer of the sphere spectrum.

III) EE-Local stable homotopy theory

To say what all this means, we recall now Bousfield localization of spectra and then indicate the tower of localizations at the Morava E-theory spectra, the “chromatic filtration”.

Bousfield localization of spectra

Definition

Let ESpecE \in Spec be a spectrum.

Say that another spectrum XSpecX \in Spec is an EE-acyclic spectrum if the smash product is zero, EX0E \wedge X \simeq 0.

Say that XX is an EE-local spectrum if every morphism YXY \longrightarrow X out of an EE-acyclic spectrum YY is homotopic to the zero morphism.

Say that a morphism f:XYf \colon X \to Y is an EE-equivalence if it becomes an equivalence after smash product with EE.

(e.g. Lurie, Lecture 20, example 4)

Proposition

For EE a spectrum, every other spectrum sits in an essentially unique homotopy cofiber sequence

G E(X)XL E(X), G_E(X) \to X \to L_E(X) \,,

where G E(X)G_E(X) is EE-acyclic, and L E(X)L_E(X) is EE-local, def. .

Here XL E(X)X \to L_E (X) is characterized by two properties

  1. L E(X)L_E(X) is EE-local;

  2. XL E(X)X \to L_E(X) is an EE-equivalence

according to def. .

(e.g. Lurie, Lecture 20, example 4)

Definition

Given ESpecE \in Spec, the natural morphisms XL EXX \longrightarrow L_E X in prop. exhibit the localization of an (infinity,1)-category called Bousfield localization at EE.

Example

For EE an E-∞ ring, every ∞-module XX over EE is EE-local, def. .

(e.g. Lurie, Lecture 20, example 5)

Example

For EE an E-∞ algebra over an E-∞ ring SS and for XX an SS-∞-module, consider the dual Cech nerve cosimplicial object

E S +1 SX:ΔSpectra. E^{\wedge_S^{\bullet+1}}\wedge_S X \;\colon\; \Delta \longrightarrow Spectra \,.

By example each term is EE-local, so that the map to the totalization

XlimE S +1 SX X \longrightarrow \underset{\leftarrow}{\lim} E^{\wedge_S^{\bullet+1}} \wedge_S X

factors through the EE-localization of XX

XL EXlimE S +1 SX. X \longrightarrow L_E X \longrightarrow \underset{\leftarrow}{\lim} E^{\wedge_S^{\bullet+1}} \wedge_S X \,.

Under suitable condition the second map here is indeed an equivalence, in which case the totalization of the dual Cech nerve exhibits the EE-localization. This happens for instance in the discussion of the Adams spectral sequence, see the examples given there.

Example

For pp \in \mathbb{N} a prime number, let

EH/p E \coloneqq H \mathbb{Z}/p\mathbb{Z}

be the corresponding Eilenberg-MacLane spectrum. Then a spectrum which corresponds to a chain complex under the stable Dold-Kan corespondence is EE-local, def. , if that chain complex has chain homology groups being [p 1]\mathbb{Z}[p^{-1}]-modules.

The EE-localization of a spectrum in this case is called p-localization.

(e.g. Lurie, Lecture 20, example 8)

Chromatic layers

Let

Definition

Write W(k)W(k) for the ring of Witt vectors. Write

RW(k)[[v 1,,v n1]] R \coloneqq W(k)[ [ v_1, \cdots, v_{n-1} ] ]

for the ring of formal power series over this ring, in n1n-1 variables; called the Lubin-Tate ring.

Theorem

The Lubin-Tate formal group f¯\overline{f} is the universal deformation of ff in that for every infinitesimal thickening AA of kk, f¯\overline{f} induces a bijection

Hom /k(R,A)Def(A) Hom_{/k}(R,A) \stackrel{\simeq}{\longrightarrow} Def(A)

between the kk-algebra-homomorphisms from RR into AA and the deformations of AA.

(e.g. Lurie 10, lect 21, theorem 5)

By the discussion there, this is Landweber exact, hence defines a cohomology theory. Therefore by the Landweber exact functor theorem there is an even periodic cohomology theory E(n) E(n)^\bullet represented by a spectrum E(n)E(n) with the property that its homotopy groups are

π (E(n))W(k)[[v 1,,v n1]][β ±1] \pi_\bullet(E(n)) \simeq W(k)[ [v_1, \cdots, v_{n-1} ] ] [ \beta^{\pm 1} ]

for β\beta of degree 2. This is called alternatively nnth Morava E-theory, or Lubin-Tate theory or Johnson-Wilson theory.

(e.g. Lurie, lect 22)

For each prime pp \in \mathbb{N} and for each natural number nn \in \mathbb{N} there is a Bousfield localization of spectra

L nL E(n), L_n \coloneqq L_{E(n)} \,,

where E(n)E(n) is the nnth Morava E-theory (for the given prime pp), called the nnth chromatic localization. These arrange into the chromatic tower which for each spectrum XX is of the form

XL nXL n1XL 0X. X \to \cdots \to L_n X \to L_{n-1} X \to \cdots \to L_0 X \,.

The homotopy fibers of each stage of the tower

M n(X)fib(L E(n)XL E(n1)(X)) M_n(X) \coloneqq fib(L_{E(n)}X \longrightarrow L_{E(n-1)}(X))

is called the nnth monochromatic layer of XX.

chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum HH \mathbb{Z}HZR-theory
0th Morava K-theoryK(0)K(0)
1complex K-theorycomplex K-theory spectrum KUKUKR-theory
first Morava K-theoryK(1)K(1)
first Morava E-theoryE(1)E(1)
2elliptic cohomologyelliptic spectrum Ell EEll_E
second Morava K-theoryK(2)K(2)
second Morava E-theoryE(2)E(2)
algebraic K-theory of KUK(KU)K(KU)
3 …10K3 cohomologyK3 spectrum
nnnnth Morava K-theoryK(n)K(n)
nnth Morava E-theoryE(n)E(n)BPR-theory
n+1n+1algebraic K-theory applied to chrom. level nnK(E n)K(E_n) (red-shift conjecture)
\inftycomplex cobordism cohomologyMUMR-theory

IV) Adams spectral sequence for EE-local homotopy groups

Summing up the above, we need a means to compute homotopy groups of EE-localized spectra. In (Lurie, Higher Algebra, section 1.2.2) is given a general spectral sequence of a filtered stable homotopy type which computes homotopy groups of spectra, and in (Lurie 10, lectures 8 and 9) is discussed that the totalization of the coskeleton filtration on the dual Cech nerve of an E-∞ algebra yields the EE-localization. Taken together this is just what we need… and this is the general EE-Adams spectral sequence. We follow the nice exposition in (Wilson 13).

First we recall

for the general case of filtered objects in suitable stable (∞,1)-categories. Then we consider the specialization of that to the

In conclusion this yields for each suitable E-∞ algebra EE over SS and SS-∞-module XX a spectral sequence converging to the homotopy groups of the EE-localization of XX, and this is

Spectral sequences for homotopy groups of filtered spectra

We discuss the spectral sequence of a filtered stable homotopy type.

Let throughout 𝒞\mathcal{C} be a stable (∞,1)-category, 𝒜\mathcal{A} an abelian category, and π:𝒞𝒜\pi \;\colon\; \mathcal{C}\longrightarrow \mathcal{A} a homological functor on 𝒞\mathcal{C}, i.e., a functor that transforms every cofiber sequence

XYZΣX X\to Y\to Z\to \Sigma X

in 𝒞\mathcal{C} into a long exact sequence

π(X)π(Y)π(Z)π(ΣX) \dots \to \pi(X)\to \pi(Y)\to \pi(Z)\to \pi(\Sigma X) \to \dots

in 𝒜\mathcal{A}. We write π n=πΣ n\pi_n=\pi\circ \Sigma^{-n}.

Example
  • 𝒞\mathcal{C} is arbitrary, 𝒜\mathcal{A} is the category of abelian groups and π\pi is taking the 0th homotopy group π 0𝒞(S,)\pi_0 \mathcal{C}(S,-) of the mapping spectrum out of some object S𝒞S\in\mathcal{C}

  • 𝒞\mathcal{C} is equipped with a t-structure, 𝒜\mathcal{A} is the heart of the t-structure, and π\pi is the canonical functor.

  • 𝒞=D(𝒜)\mathcal{C} = D(\mathcal{A}) is the derived category of the abelian category 𝒜\mathcal{A} and π=H 0\pi=H_0 is the degree-0 chain homology functor.

  • Any of the above with 𝒞\mathcal{C} and 𝒜\mathcal{A} replaced by their opposite categories.

Definition

A filtered object in an (∞,1)-category in 𝒞\mathcal{C} is simply a sequential diagram X:(,<)𝒞X \colon (\mathbb{Z}, \lt) \to \mathcal{C}

X n1X nX n+1. \cdots X_{n-1} \to X_n \to X_{n+1} \to \cdots \,.

This appears as (Higher Algebra, def. 1.2.2.9).

We will take the view that the object being filtered is the homotopy limit

Xlim nX n. X \coloneqq \underset{\leftarrow}{\lim}_n X_n.

We could also consider the sequential diagram as a filtering of its homotopy colimit, but this is really an equivalent point of view since we can replace 𝒞\mathcal{C} by 𝒞 op\mathcal{C}^{op}.

Definition

Let II be a linearly ordered set. An II-chain complex in a stable (∞,1)-category 𝒞\mathcal{C} is an (∞,1)-functor

F:I Δ[1]𝒞 F \;\colon\; I^{\Delta[1]} \longrightarrow \mathcal{C}

such that

  1. for each nIn \in I, F(n,n)0F(n,n) \simeq 0 is the zero object;

  2. for all ijki \leq j \leq k the induced diagram

    F(i,j) F(i,k) F(j,j) F(j,k) \array{ F(i,j) &\longrightarrow& F(i,k) \\ \downarrow && \downarrow \\ F(j,j) &\longrightarrow& F(j,k) }

    is a homotopy pullback square.

This is Higher Algebra, def. 1.2.2.2.

Remark

Given a \mathbb{Z}-chain complex FF in 𝒞\mathcal{C} as in def. , setting

C nΣ nF(n,n+1) C_n \coloneqq \Sigma^{-n} F(n,n+1)

and defining a differential induced from the connecting homomorphisms of the defining homotopy fiber sequences

F(n1,n)F(n1,n+1)F(n,n+1) F(n-1,n) \to F(n-1, n+1) \to F(n,n+1)

yields an ordinary chain complex C C_\bullet in the homotopy category.

(Higher Algebra, remark 1.2.2.3)

Proposition

Consider the inclusion of posets

(,)({},) Δ[1] (\mathbb{Z}, \leq) \to (\mathbb{Z}\cup \{\infty\}, \leq)^{\Delta[1]}

given by

n(n,). n \mapsto (n,\infty) \,.

The induced (∞,1)-functor

Func(({},) Δ[1],𝒞)Func((,),𝒞) Func((\mathbb{Z}\cup \{\infty\}, \leq)^{\Delta[1]} , \mathcal{C}) \longrightarrow Func((\mathbb{Z}, \leq), \mathcal{C})

restricts to an equivalence between the (∞,1)-category of {}\mathbb{Z}\cup \{\infty\}-chain complexes in 𝒞\mathcal{C} (def. ) and that of generalized filtered objects in 𝒞\mathcal{C} (def. ).

This is Higher Algebra, lemma 1.2.2.4. The inverse functor can be described informally as follows: given a filtered object X X_\bullet, the associated chain complex X(,)X(\bullet,\bullet) is given by

X(n,n+r)=fib(X nX n+r). X(n, n+r) = \operatorname{fib}(X_n\to X_{n+r}).
Definition

Let X X_\bullet be a filtered object in the sense of def. . Write X(,)X(\bullet,\bullet) for the corresponding chain complex, according to prop. .

Then for all ijki \leq j \leq k there is a long exact sequence of homotopy groups in 𝒜\mathcal{A} of the form

π nX(i,j)π nX(i,k)π nX(j,k)π n1X(i,j). \cdots \to \pi_n X(i,j) \to \pi_n X(i,k) \to \pi_n X(j,k) \to \pi_{n-1}X(i,j) \to \cdots \,.

Define then for p,qp,q \in \mathbb{Z} and r1r \geq 1 the object E p,q rE^r_{p,q} by the canonical epi-mono factorization

π pX(qr+1,q+1)E p,q rπ pX(q,q+r) \pi_{p} X(q-r+1,q+1) \twoheadrightarrow E^r_{p,q} \hookrightarrow \pi_{p} X(q, q+r)

in the abelian category 𝒜\mathcal{A}, and define the differential

d r:E p,q rE p1,qr r d^r \;\colon\; E_{p,q}^r \to E_{p-1, q-r}^r

to be the restriction of the connecting homomorphism

π pX(q,q+r)π p1X(qr,q) \pi_{p} X(q,q+r) \to \pi_{p-1} X(q-r, q)

from the above long exact sequence (with i=qri=q-r, j=qj=q, and k=q+rk=q+r).

(Higher Algebra, construction 1.2.2.6)

Proposition

d rd r=0d^r\circ d^r = 0 and there are natural (in X X_\bullet) isomorphisms

E r+1ker(d r)/im(d r). E^{r+1}\cong \operatorname{ker}(d^r)/\operatorname{im}(d^r).

Thus, {E *,* r} r1\{E^r_{*,*}\}_{r\geq 1} is a bigraded spectral sequence in the abelian category 𝒜\mathcal{A}, functorial in the filtered object X X_\bullet, with

E p,q 1=π pfib(X qX q+1),d r:E p,q rE p1,qr r. E^1_{p,q} = \pi_p \operatorname{fib}(X_q\to X_{q+1}), \qquad d^r: E^r_{p,q}\to E^r_{p-1,q-r}.

(Higher Algebra, prop. 1.2.2.7)

If sequential limits and sequential colimits exist in 𝒜\mathcal{A}, we can form the limiting term E *,* E^\infty_{*,*} of this spectral sequence.

On the other hand, the graded object π (X)\pi_\bullet (X) admits a filtration by

F qπ p(X)=ker(π p(X)π p(X q)) F_q \pi_p (X) = \operatorname{ker}(\pi_p (X)\to \pi_p(X_q))

and we would like to compare E *,* E^\infty_{*,*} with the associated graded of this filtration. We say that

Definition

The spectral sequence converges weakly if there is a canonical isomorphism

E p,q F qπ p(X)/F q1π p(X) E^\infty_{p,q} \cong F_q\pi_p(X)/ F_{q-1}\pi_p(X)

for every p,qp,q\in\mathbb{Z}.

We say that the spectral sequence converges strongly if it converges weakly and if, in addition, the filtration F π p(X)F_\bullet\pi_p(X) is complete on both sides.

Remark

The meaning of the word canonical in def. is somewhat subtle since, in general, there is no map from one side to the other. However, there always exists a canonical relation between the two, and we ask that this relation be an isomorphism (see Hilton-Stammbach, VIII.7).

Proposition

Let 𝒞\mathcal{C} be a stable (∞,1)-category and let π:𝒞𝒜\pi:\mathcal{C}\to\mathcal{A} be a homological functor where 𝒜\mathcal{A} is an abelian category which admits sequential limits. Let X X_\bullet be a filtered object in 𝒞\mathcal{C} such that limX \underset{\leftarrow}{\lim} X_\bullet exists. Suppose further that:

  1. For every nn, the diagram rfib(X nrX n)r\mapsto \operatorname{fib}(X_{n-r}\to X_n) has a limit in 𝒞\mathcal{C} and that limit is preserved by π\pi.
  2. For every nn, π n(X r)=0\pi_n(X_r)=0 for r0r\gg 0.

Then the spectral sequence {E *,* r} r1\{E^r_{*,*}\}_{r\geq 1} in 𝒜\mathcal{A} converges strongly (def. ). We write:

E p,q 1=π pfib(X qX q+1)π p(limX ) E_{p,q}^1 = \pi_{p} \operatorname{fib}(X_q\to X_{q+1}) \Rightarrow \pi_{p} (\underset{\leftarrow}{\lim} X_\bullet)

There is also a dual statement in which limits are replaced by colimits, but it is in fact a special case of the proposition with π\pi replaced by π op\pi^{op}. A proof of this proposition (in dual form) is given in (Higher Algebra, prop. 1.2.2.14). Review is in (Wilson 13, theorem 1.2.1).

For the traditional statement in the category of chain complexes see at spectral sequence of a filtered complex.

Plenty of types of spectral sequences turn out to be special cases of this general construction.

tower diagram/filteringspectral sequence of a filtered stable homotopy type
filtered chain complexspectral sequence of a filtered complex
Postnikov towerAtiyah-Hirzebruch spectral sequence
chromatic towerchromatic spectral sequence
skeleta of simplicial objectspectral sequence of a simplicial stable homotopy type
skeleta of Sweedler coring of E-∞ algebraAdams spectral sequence
filtration by support
slice filtrationslice spectral sequence

Canonical cosimplicial resolution of E E_\infty-algebras

We discuss now the special case of coskeletally filtered totalizations coming from the canonical cosimplicial objects induced from E-∞ algebras (dual Cech nerves/Sweedler corings/Amitsur complexes).

In this form this appears as (Lurie 10, theorem 2). A review is in (Wilson 13, 1.3). For the analog of this in the traditional formulation see (Ravenel, ch. 3, prop. 3.1.2).

Definition

Given an cosimplicial object in an (∞,1)-category with (∞,1)-colimits

Y:Δ𝒞 Y \;\colon\; \Delta \longrightarrow \mathcal{C}

its totalization TotYlim nY nTot Y \simeq \underset{\leftarrow}{\lim}_n Y_n is filtered, def. , by the totalizations of its coskeleta

TotYTot(cosk 2Y)Tot(cosk 1Y)Tot(cosk 0Y)0. Tot Y \to \cdots \to Tot (cosk_2 Y) \to Tot (cosk_1 Y) \to Tot (cosk_0 Y) \to 0 \,.
Definition

The filtration spectral sequence, prop. , applied to the filtration of a totalization by coskeleta as in def. , we call the spectral sequence of a simplicial stable homotopy type.

(Higher Algebra, prop. 1.2.4.5)

Proposition

The spectral sequence of a simplicial stable homotopy type has as first page/E 1E_1-term the cohomology groups of the Moore complex associated with the cosimplicial objects of homotopy groups

E 2 p,q=H p(π q(Tot(cosk (Y))))π pqTot(Y). E_2^{p,q} = H^p(\pi_q(Tot (cosk_\bullet(Y)))) \Rightarrow \pi_{p-q} Tot(Y) \,.

By the discussion at ∞-Dold-Kan correspondence and spectral sequence of a filtered stable homotopy type. This appears as (Higher Algebra, remark 1.2.4.4). Review is around (Wilson 13, theorem 1.2.4).

Definition

Let SS be an E-∞ ring and let EE be an E-∞ algebra over SS, hence an E-∞ ring equipped with a homomorphism

SE. S \longrightarrow E \,.

The canonical cosimplicial object associated to this (the \infty-Cech nerve/Sweedler coring/Amitsur complex) is that given by the iterated smash product/tensor product over SS:

E S +1:Δ𝒞. E^{\wedge^{\bullet+1}_S} \;\colon\; \Delta \to \mathcal{C} \,.

More generally, for XX an SS-∞-module, the canonical cosimplicial object is

E S +1 SX:Δ𝒞. E^{\wedge^{\bullet+1}_S}\wedge_S X \;\colon\; \Delta \to \mathcal{C} \,.
Proposition

If EE is such that the self-generalized homology E (E)π (E SE)E_\bullet(E) \coloneqq \pi_\bullet(E \wedge_S E) (the dual EE-Steenrod operations) is such that as a module over E π (E)E_\bullet \coloneqq \pi_\bullet(E) it is a flat module, then there is a natural equivalence

π (E S n+1 SX)E (E S n) E E (X). \pi_\bullet \left( E^{\wedge^{n+1}_S} \wedge_S X \right) \simeq E_\bullet(E^{\wedge^n_S}) \otimes_{E_\bullet} E_\bullet(X) \,.

Reviewed for instance as (Wilson 13, prop. 1.3.1).

Remark

This makes (E ,E (E))(E_\bullet, E_\bullet(E)) be the commutative Hopf algebroid formed by the EE-Steenrod algebra. See there for more on this.

Example

The condition in prop. is satisfied for

It is NOT satisfied for

Remark

Under good conditions (…), π \pi_\bullet of the canonical cosimplicial object provides a resolution of comodule tensor product and hence computes the Ext-groups over the commutative Hopf algebroid:

H p(π q(Tot(cosk (E S +1 SX))))Ext E (E) p(Σ qE ,E (X)). H^p(\pi_q(Tot(cosk_\bullet(E^{\wedge^{\bullet+1}_S } \wedge_S X)))) \simeq Ext^p_{E_\bullet(E)}(\Sigma^q E_\bullet, E_\bullet(X)) \,.

(…)

Remark

There is a canonical map

L EXlim n(E S n+1 SX) L_E X \stackrel{}{\longrightarrow} \underset{\leftarrow}{\lim}_n (E^{\wedge^{n+1}_S}\wedge_S X)

from the EE-Bousfield localization of spectra of XX into the totalization.

(Lurie 10, lecture 30, prop. 1)

We consider now conditions for this morphism to be an equivalence.

Definition

For RR a ring, its core cRc R is the equalizer in

cRRRR. c R \longrightarrow R \stackrel{\longrightarrow}{\longrightarrow} R \otimes R \,.
Proposition

Let EE be a connective E-∞ ring such that the core or π 0(E)\pi_0(E), def. is either of

  • the localization of the integers at a set JJ of primes, cπ 0(E)matbbZ[J 1]c \pi_0(E) \simeq \matbb{Z}[J^{-1}];

  • n\mathbb{Z}_n for n2n \geq 2.

Then the map in remark is an equivalence

L EXlim n(E S n+1 SX). L_E X \stackrel{\simeq}{\longrightarrow} \underset{\leftarrow}{\lim}_n (E^{\wedge^{n+1}_S}\wedge_S X) \,.

(Bousfield 79, Lurie 10, lecture 30, prop. 3, Lurie 10, lecture 31,).

The EE-Adams-Novikov spectral sequence

Summing this up yields the general EE-Adams(-Novikov) spectral sequence

Corollary

Let EE a connective E-∞ ring that satisfies the conditions of prop. . Then by prop. and prop. there is a strongly convergent multiplicative spectral sequence

E p,qπ qpL cπ 0EX E^{p,q}_\bullet \Rightarrow \pi_{q-p} L_{c \pi_0 E} X

converging to the homotopy groups of the cπ 0(E) c \pi_0(E)-localization of XX. If moreover the dual EE-Steenrod algebra E (E)E_\bullet(E) is flat as a module over E E_\bullet, then, by prop. and remark , the E 1E_1-term of this spectral sequence is given by the Ext-groups over the EE-Steenrod Hopf algebroid.

E p,q=Ext E (E) p(Σ qE ,E X). E^{p,q}_\bullet = Ext^p_{E_\bullet(E)}(\Sigma^q E_\bullet, E_\bullet X) \,.

References

An introduction to the chromatic perspective on the homotopy groups of spheres and the image of JJ is in:

The bulk of the basic constructions is in

Recent surveys of the modern picture are in

and of relevance for the above discussion are particularly the following contributions there

  • Ben Knudsen, First chromatic layer of the sphere spectrum = homotopy of the K(1)K(1)-local sphere, talk at 2013 Pre-Talbot Seminar (pdf)

Loads of details for computations in the Adams spectral sequence are in

Last revised on December 4, 2013 at 01:31:53. See the history of this page for a list of all contributions to it.