nLab J-homomorphism and chromatic homotopy

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 $E$-Adams spectral sequence.

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

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

Remark

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

$$O(n) \times S^n \longrightarrow S^n$$

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

For definiteness we distinguish in the following notationally between

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

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

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

Definition

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

$$H(n) \coloneqq Aut_{\infty Grpd^{\ast/}}(\Pi(S^n)) \,.$$

Remark

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

Definition

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

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

induces a homomorphism of ∞-groups of the form

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

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

$$\pi_i(O(n)) \longrightarrow \pi_i(\Omega^n S^n) \simeq \pi_{n+i}(S^n) \,.$$

Remark

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

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

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

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

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

Therefore one can take the direct limit over $n$:

Definition

By remark there is induced a homomorphism

$$J_i \;\colon\; \pi_\bullet(O) \longrightarrow \pi_\bullet(\mathbb{S})$$

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

$$B J \;\colon\; B O \longrightarrow B GL_1(\mathbb{S}) = B H \,.$$

Remark

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

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

Definition

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

It is immediate that:

Proposition

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

$n\;mod\; 8$	0	1	2	3	4	5	6	7
$\pi_n(O)$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	0	$\mathbb{Z}$	0	0	0	$\mathbb{Z}$

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

$n\;mod\; 8$	0	1	2	3	4	5	6	7
${\vert\pi_n(O)\vert}$	2	2	1	$\infty$	1	1	1	$\infty$

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

$n\;mod\; 8$	$n = 0$	0	1	2	3	4	5	6	7
$\pi_n(S O)$	0	$\mathbb{Z}_2$	$\mathbb{Z}_2$	0	$\mathbb{Z}$	0	0	0	$\mathbb{Z}$

$n\;mod\; 8$	$n = 0$	0	1	2	3	4	5	6	7
${\vert\pi_n(S O)\vert}$	1	2	2	1	$\infty$	1	1	1	$\infty$

Theorem

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

Moreover,

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

• for $n = 3\; mod\; 8$ and $n = 7 \; mod \; 8$ hence for $n = 4 k -1$, the order of the image is equal to the denominator of $B_{2k}/4k$ in its reduced form, where $B_{2k}$ 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 $J(\pi_{4n-1}(S O))$ 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 $J$ is in (Ravenel, chapter 5, section 3).

Remark

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

k	1	2	3	4	5	6	7	8	9	10
$\vert J(\pi_{4k-1}(O))\vert$	24	240	504	480	264	65,520	24	16,320	28,728	13,200

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

Remark

Therefore we have in low degree the following situation

	$n$	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Whitehead tower of orthogonal group		orientation	spin group		string group				fivebrane group				ninebrane group
higher versions		special orthogonal group	spin group		string 2-group				fivebrane 6-group				ninebrane 10-group
homotopy groups of stable orthogonal group	$\pi_n(O)$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	0	$\mathbb{Z}$	0	0	0	$\mathbb{Z}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	0	$\mathbb{Z}$	0	0	0	$\mathbb{Z}$	$\mathbb{Z}_2$
stable homotopy groups of spheres	$\pi_n(\mathbb{S})$	$\mathbb{Z}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	$\mathbb{Z}_{24}$	0	0	$\mathbb{Z}_2$	$\mathbb{Z}_{240}$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$	$\mathbb{Z}_6$	$\mathbb{Z}_{504}$	0	$\mathbb{Z}_3$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2$	$\mathbb{Z}_{480} \oplus \mathbb{Z}_2$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2$
image of J-homomorphism	$im(\pi_n(J))$	0	$\mathbb{Z}_2$	0	$\mathbb{Z}_{24}$	0	0	0	$\mathbb{Z}_{240}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	0	$\mathbb{Z}_{504}$	0	0	0	$\mathbb{Z}_{480}$	$\mathbb{Z}_2$

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

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

$p = 3$-primary component

$p = 5$-primary component

We illustrate how to read these tables:

Example

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

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

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

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

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

Similarly the group $\pi_9(\mathbb{S}) \simeq \mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$ corresponds to the three unconnected dots above $n = 9$ 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 $E(1)$ for the first Morava E-theory spectrum at given prime number $p$. Write $L_{E(1)}\mathbb{S}$ for the Bousfield localization of spectra of the sphere spectrum at $E(1)$.

Theorem

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

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

This appears as (Lurie 10, theorem 6)

Definition

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

$$J \;\colon\; \pi_n(O) \longrightarrow \pi_n(\mathbb{S}) \longrightarrow \pi_n(\mathbb{S}_{(p)}) \,.$$

Theorem

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

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

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

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

Corollary

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

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

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

Remark

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

III) $E$-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 $E \in Spec$ be a spectrum.

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

Say that $X$ is an $E$-local spectrum if every morphism $Y \longrightarrow X$ out of an $E$-acyclic spectrum $Y$ is homotopic to the zero morphism.

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

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

Proposition

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

$$G_E(X) \to X \to L_E(X) \,,$$

where $G_E(X)$ is $E$-acyclic, and $L_E(X)$ is $E$-local, def. .

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

1. $L_E(X)$ is $E$-local;

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

according to def. .

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

Definition

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

Example

For $E$ an E-∞ ring, every ∞-module $X$ over $E$ is $E$-local, def. .

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

Example

For $E$ an E-∞ algebra over an E-∞ ring $S$ and for $X$ an $S$-∞-module, consider the dual Cech nerve cosimplicial object

$$E^{\wedge_S^{\bullet+1}}\wedge_S X \;\colon\; \Delta \longrightarrow Spectra \,.$$

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

$$X \longrightarrow \underset{\leftarrow}{\lim} E^{\wedge_S^{\bullet+1}} \wedge_S X$$

factors through the $E$-localization of $X$

$$X \longrightarrow L_E X \longrightarrow \underset{\leftarrow}{\lim} E^{\wedge_S^{\bullet+1}} \wedge_S X \,.$$

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

Example

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

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

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

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

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

Chromatic layers

Let

• $k$ be a perfect field of characteristic $p$;

• $f$ be a formal group of height $n$ over $k$.

Definition

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

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

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

Theorem

The Lubin-Tate formal group $\overline{f}$ is the universal deformation of $f$ in that for every infinitesimal thickening $A$ of $k$, $\overline{f}$ induces a bijection

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

between the $k$-algebra-homomorphisms from $R$ into $A$ and the deformations of $A$.

(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 $E(n)^\bullet$ represented by a spectrum $E(n)$ with the property that its homotopy groups are

$$\pi_\bullet(E(n)) \simeq W(k)[ [v_1, \cdots, v_{n-1} ] ] [ \beta^{\pm 1} ]$$

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

(e.g. Lurie, lect 22)

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

$$L_n \coloneqq L_{E(n)} \,,$$

where $E(n)$ is the $n$th Morava E-theory (for the given prime $p$), called the $n$th chromatic localization. These arrange into the chromatic tower which for each spectrum $X$ is of the form

$$X \to \cdots \to L_n X \to L_{n-1} X \to \cdots \to L_0 X \,.$$

The homotopy fibers of each stage of the tower

$$M_n(X) \coloneqq fib(L_{E(n)}X \longrightarrow L_{E(n-1)}(X))$$

is called the $n$th monochromatic layer of $X$.

chromatic homotopy theory

chromatic level	complex oriented cohomology theory	E-∞ ring/A-∞ ring	real oriented cohomology theory
0	ordinary cohomology	Eilenberg-MacLane spectrum $H \mathbb{Z}$	HZR-theory
	0th Morava K-theory	$K(0)$
1	complex K-theory	complex K-theory spectrum $KU$	KR-theory
	first Morava K-theory	$K(1)$
	first Morava E-theory	$E(1)$
2	elliptic cohomology	elliptic spectrum $Ell_E$
	second Morava K-theory	$K(2)$
	second Morava E-theory	$E(2)$
	algebraic K-theory of KU	$K(KU)$
3 …10	K3 cohomology	K3 spectrum
$n$	$n$th Morava K-theory	$K(n)$
	$n$th Morava E-theory	$E(n)$	BPR-theory
$n+1$	algebraic K-theory applied to chrom. level $n$	$K(E_n)$ (red-shift conjecture)
$\infty$	complex cobordism cohomology	MU	MR-theory

IV) Adams spectral sequence for $E$-local homotopy groups

Summing up the above, we need a means to compute homotopy groups of $E$-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 $E$-localization. Taken together this is just what we need… and this is the general $E$-Adams spectral sequence. We follow the nice exposition in (Wilson 13).

First we recall

• Spectral sequences computing homotopy groups of filtered objects

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

• Canonical cosimplicial resolutions of E-∞ algebras.

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

• The E-Adams-Novikov spectral sequence.

Spectral sequences for homotopy groups of filtered spectra

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

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

$$X\to Y\to Z\to \Sigma X$$

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

$$\dots \to \pi(X)\to \pi(Y)\to \pi(Z)\to \pi(\Sigma X) \to \dots$$

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

Example

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

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

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

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

Definition

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

$$\cdots X_{n-1} \to X_n \to X_{n+1} \to \cdots \,.$$

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

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

$$X \coloneqq \underset{\leftarrow}{\lim}_n X_n.$$

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 $\mathcal{C}$ by $\mathcal{C}^{op}$.

Definition

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

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

such that

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

2. for all $i \leq j \leq k$ the induced diagram

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

   is a homotopy pullback square.

This is Higher Algebra, def. 1.2.2.2.

Remark

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

$$C_n \coloneqq \Sigma^{-n} F(n,n+1)$$

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

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

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

(Higher Algebra, remark 1.2.2.3)

Proposition

Consider the inclusion of posets

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

given by

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

The induced (∞,1)-functor

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

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

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

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

Definition

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

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

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

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

$$\pi_{p} X(q-r+1,q+1) \twoheadrightarrow E^r_{p,q} \hookrightarrow \pi_{p} X(q, q+r)$$

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

$$d^r \;\colon\; E_{p,q}^r \to E_{p-1, q-r}^r$$

to be the restriction of the connecting homomorphism

$$\pi_{p} X(q,q+r) \to \pi_{p-1} X(q-r, q)$$

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

(Higher Algebra, construction 1.2.2.6)

Proposition

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

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

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

$$E^1_{p,q} = \pi_p \operatorname{fib}(X_q\to X_{q+1}), \qquad d^r: E^r_{p,q}\to E^r_{p-1,q-r}.$$

(Higher Algebra, prop. 1.2.2.7)

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

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

$$F_q \pi_p (X) = \operatorname{ker}(\pi_p (X)\to \pi_p(X_q))$$

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

Definition

The spectral sequence converges weakly if there is a canonical isomorphism

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

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

We say that the spectral sequence converges strongly if it converges weakly and if, in addition, the filtration $F_\bullet\pi_p(X)$ 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 $\mathcal{C}$ be a stable (∞,1)-category and let $\pi:\mathcal{C}\to\mathcal{A}$ be a homological functor where $\mathcal{A}$ is an abelian category which admits sequential limits. Let $X_\bullet$ be a filtered object in $\mathcal{C}$ such that $\underset{\leftarrow}{\lim} X_\bullet$ exists. Suppose further that:

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

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

$$E_{p,q}^1 = \pi_{p} \operatorname{fib}(X_q\to X_{q+1}) \Rightarrow \pi_{p} (\underset{\leftarrow}{\lim} X_\bullet)$$

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 $\pi$ replaced by $\pi^{op}$. 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/filtering	spectral sequence of a filtered stable homotopy type
filtered chain complex	spectral sequence of a filtered complex
Postnikov tower	Atiyah-Hirzebruch spectral sequence
chromatic tower	chromatic spectral sequence
skeleta of simplicial object	spectral sequence of a simplicial stable homotopy type
skeleta of Sweedler coring of E-∞ algebra	Adams spectral sequence
filtration by support	…
slice filtration	slice spectral sequence

Canonical cosimplicial resolution of $E_\infty$-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

$$Y \;\colon\; \Delta \longrightarrow \mathcal{C}$$

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

$$Tot Y \to \cdots \to Tot (cosk_2 Y) \to Tot (cosk_1 Y) \to Tot (cosk_0 Y) \to 0 \,.$$

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/$E_1$-term the cohomology groups of the Moore complex associated with the cosimplicial objects of homotopy groups

$$E_2^{p,q} = H^p(\pi_q(Tot (cosk_\bullet(Y)))) \Rightarrow \pi_{p-q} Tot(Y) \,.$$

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 $S$ be an E-∞ ring and let $E$ be an E-∞ algebra over $S$, hence an E-∞ ring equipped with a homomorphism

$$S \longrightarrow E \,.$$

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 $S$:

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

More generally, for $X$ an $S$-∞-module, the canonical cosimplicial object is

$$E^{\wedge^{\bullet+1}_S}\wedge_S X \;\colon\; \Delta \to \mathcal{C} \,.$$

Proposition

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

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

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

Remark

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

Example

The condition in prop. is satisfied for

• $E = H \mathbb{F}_p$ an Eilenberg-MacLane spectrum with $mod\;p$ coefficients;

• $E = B P$ the Brown-Peterson spectrum;

• $E = MU$ the complec cobordism spectrum.

It is NOT satisfied for

• $E = H \mathbb{Z}$ the Eilenberg-MacLane spectrum for integers coefficients;

• $E = M S U$.

Remark

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

$$H^p(\pi_q(Tot(cosk_\bullet(E^{\wedge^{\bullet+1}_S } \wedge_S X)))) \simeq Ext^p_{E_\bullet(E)}(\Sigma^q E_\bullet, E_\bullet(X)) \,.$$

(…)

Remark

There is a canonical map

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

from the $E$-Bousfield localization of spectra of $X$ into the totalization.

(Lurie 10, lecture 30, prop. 1)

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

Definition

For $R$ a ring, its core $c R$ is the equalizer in

$$c R \longrightarrow R \stackrel{\longrightarrow}{\longrightarrow} R \otimes R \,.$$

Proposition

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

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

• $\mathbb{Z}_n$ for $n \geq 2$.

Then the map in remark is an equivalence

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

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

The $E$-Adams-Novikov spectral sequence

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

Corollary

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

$$E^{p,q}_\bullet \Rightarrow \pi_{q-p} L_{c \pi_0 E} X$$

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

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

References

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

• Mark Mahowald, Doug Ravenel, Towards a Global Understanding of the Homotopy Groups of Spheres, 1998 (pdf)

The bulk of the basic constructions is in

• Jacob Lurie, Chromatic Homotopy Theory, lecture notes (2010)

• Jacob Lurie, Higher Algebra

Recent surveys of the modern picture are in

• Talbot 2013: Chromatic Homotopy Theory, 2013 Pre-Talbot Seminar Chromatic homotopy theoy

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

• Mark Behrens, section 1 of Introduction talk at Talbot 2013: Chromatic Homotopy Theory (pdf, pdf)

• Dylan WilsonSpectral Sequences from Sequences of Spectra: Towards the

  Spectrum of the Category of Spectra_ lecture at 2013 Pre-Talbot Seminar (pdf)

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

• Vitaly Lorman, Chromatic homotopy theory at height 1 and the

  image of $J$, talk atTalbot 2013: Chromatic Homotopy Theory_ (pdf)

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

• Doug Ravenel, Complex cobordism and stable homotopy groups of spheres, 1986

The construction of the image of the J-homomorphism when twisted by Dirichlet characters is in

• Ningchuan Zhang, Analogs of Dirichlet L-functions in chromatic homotopy theory, (arXiv:1910.14582)