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Adams spectral sequence

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Idea

The Adams spectral sequence (Adams 58) is a type of spectral sequences used for computations in stable homotopy theory. It computes the homotopy groups of a spectrum from its homology/cohomology, as modules/comodules over its cohomology operations. The Adams spectral sequence can be seen as a variant of the Serre spectral sequence obtained by replacing a single fibration by an “Adams resolution”.

The original Adams spectral sequence for ordinary cohomology is further refined by the Adams-Novikov spectral sequence (Novikov 67) by replacing ordinary cohomology modulo pp by complex cobordism cohomology theory or Brown-Peterson theory or the like. Generally, for EE a suitable E-infinity algebra there is a corresponding EE-Adams(-Novikov) spectral sequence whose second page is given by EE-generalized cohomology and which arises as the spectral sequence of a simplicial stable homotopy type of the cosimplicial object which is the Cech nerve/Sweedler coring/Amitsur complex of EE. As such the Adams spectral sequence is an analog in stable homotopy theory of the Bousfield-Kan homotopy spectral sequence.

Working with the Adams spectral sequence tends to be fairly involved, as is clear from the subtlety of the results it computes (notably stable homotopy groups of spheres) and as witnessed by the fact that one uses further spectral sequences just to compute the low pages of the Adams spectral sequence, e.g. the May spectral sequence and the chromatic spectral sequence.

A neat conceptual picture of what happens in the Adams spectral sequence has emerged long after its conception with the arrival of higher algebra in stable infinity-category theory. A nice, brief, illuminating modern (and funny) account of this is in (Wilson 13), further details are in (Lurie 10).

Motivation from Hurewicz theorem and Serre spectral sequence

The Adams spectral sequence may be motivated from the strategy to compute homotopy groups from cohomology groups by subsequently applying the Hurewicz theorem to compute the lowest-degree non-trivial homotopy group from the corresponding cohomology group, then co-killing that by forming its homotopy fiber, finally applying the Serre spectral sequence to identify the next lowest non-trivial cohomology group of that fiber, and then iterating this process. The Adams spectral sequence arises when in this kind of strategy instead of co-killing only the lowest lying cohomology group, one at a time, one co-kills all nontrivial cohomology groups, then forms the corresponding homotopy fiber and so on.

This was apparently historically the way that John Adams indeed proceeded from Jean-Pierre Serre’s approach and this is still a good motivation for the whoe construction, a nice exposition is in (Wilson 13, 1.1).

We now say this again in more detail.

Given nn \in \mathbb{N}, consider the probem of computing the homotopy groups π k(S n)mod2\pi_k(S^n) \;mod \;2 of the nn-sphere S nS^n. For knk \leq n this is clear: first for k}ltnk \}lt n they all vanish, and second for k=nk = n we have, by the very nature of Eilenberg-MacLane spaces K( 2,n)K(\mathbb{Z}_2, n), that the ordinary cohomology is

H n(S n, 2)[S n,K( 2,n)]π n(K( 2,n)) 2 H^n(S^n, \mathbb{Z}_2) \simeq [S^n, K(\mathbb{Z}_2,n)] \simeq \pi_n(K(\mathbb{Z}_2,n)) \simeq \mathbb{Z}_2

so that by the Hurewicz theorem it follows that also

π n(S n)mod2 2. \pi_n(S^n) \;mod\;2 \;\simeq \mathbb{Z}_2 \,.

The Hurewicz theorem does not say anything beyong the first non-vanishing cohomology group, but so to apply it again we can move up one step in the Whitehead tower of S nS^n and hence consider the homotopy fiber

F 1 S n c 1 K( 2,n) \array{ F_1 \\ \downarrow \\ S^n &\stackrel{c_1}{\longrightarrow}& K(\mathbb{Z}_2,n) }

of the generator [c 1]=1π n(S n) 2[c_1] = 1 \in \pi_n(S^n) \simeq \mathbb{Z}_2.

To apply the Hurewicz theorem to that fiber we need to know its lowest non-trivial cohomology group again, and this is computed via the Serre spectral sequence applied to this fiber sequence.

From here on the process repeats, and one moves higher through the Whitehead tower of S nS^n

F 1 c 2 K( 2,n+1) S n c 1 K( 2,n). \array{ \vdots \\ \downarrow \\ F_1 &\stackrel{c_2}{\longrightarrow}& K(\mathbb{Z}_2, n+1) \\ \downarrow \\ S^n &\stackrel{c_1}{\longrightarrow}& K(\mathbb{Z}_2,n) } \,.

The Adams spectral sequence arises from this strategy by co-killing not just the first non-trivial cohomology group at each stage, but all nontrivial cohomology groups at a given stage.

This is done in stable homotopy theory, so let now XX be a spectrum (for instance the sphere spectrum X=𝕊X = \mathbb{S} if we still with the computation of the stable homotopy groups of spheres). Write H𝔽 2H \mathbb{F}_2 for the Eilenberg-MacLane spectrum for ordinary cohomology with coefficients in 2\mathbb{Z}_2, so that an element in cohomology

[c]H n(X) [c] \in H^n(X)

is represented by the homotopy class of a homomorphism of spectra of the form

c:XΣ nH𝔽 2 c \;\colon\; X \longrightarrow \Sigma^n H\mathbb{F}_2

(a cocycle), where “Σ\Sigma” denotes suspension, as usual.

If XX is a finite spectrum then there is a finite II of non-trivial cohomology classes like this, and a choice of cocycles c ic_i for each of them gives a single map

f 0(c i) IXK 0 iIΣ n iH𝔽 2 f_0 \coloneqq (c_i)_I \;\; X \longrightarrow K_0 \coloneqq \wedge_{i \in I} \Sigma^{n_i}H \mathbb{F}_2

into a generalized Eilenberg-MacLane spectrum. As before, this map classifies its homotopy fiber

F 1 X f 0 K 0 \array{ F_1 \\ \downarrow \\ X &\stackrel{f_0}{\longrightarrow}& K_0 }

which may be thought of as encoding all information about XX beyond its cohomology groups. Iterating this process gives the corresponding analog of the Whitehead tower, called the Adams resolution of XX:

F 2 f 2 K 2 F 1 f 1 K 1 X f 0 K 0. \array{ \vdots \\ \downarrow \\ F_2 &\stackrel{f_2}{\longrightarrow}& K_2 \\ \downarrow \\ F_1 &\stackrel{f_1}{\longrightarrow}& K_1 \\ \downarrow \\ X &\stackrel{f_0}{\longrightarrow}& K_0 } \,.

The Adams spectral sequence is that induced by the exact couple obtained by applying π \pi_\bullet to this Adams resolution.

We now say this more in detail.

The long exact sequences of homotopy groups for all the homotopy fibers in this diagram arrange into a diagram of the form

π (F 2) π (f 2) π (K 2) 2 π (F 1) π (f 1) π (K 1) 1 π (X) π (f 0) π (K 0), \array{ \vdots \\ \downarrow & \nwarrow \\ \pi_\bullet(F_2) &\stackrel{\pi_\bullet(f_2)}{\longrightarrow}& \pi_\bullet(K_2) \\ \downarrow & \nwarrow^{\mathrlap{\partial_2}} \\ \pi_\bullet(F_1) &\stackrel{\pi_\bullet(f_1)}{\longrightarrow}& \pi_\bullet(K_1) \\ \downarrow & \nwarrow^{\mathrlap{\partial_1}} \\ \pi_\bullet(X) &\stackrel{\pi_\bullet(f_0)}{\longrightarrow}& \pi_\bullet(K_0) } \,,

where the diagonal maps are the connecting homomorphisms and hence decrease degree in π \pi_\bullet by one. The idea now is to compute the homotopy groups of XX from the decomposed information in this diagram as follows.

First, by construction the homotopy groups π (K s)\pi_\bullet(K_s) are known, therefore we can identify elements

σπ (X) \sigma \in \pi_\bullet(X)

if they come from elements

σ sπ (X s) \sigma_s \in \pi_\bullet(X_s)

whose image

π (f s)(σ s)π (K s) \pi_\bullet(f_s)(\sigma_s) \in \pi_\bullet(K_s)

we understand. So the task is to understand the image of π (f s)\pi_\bullet(f_s) in π (K s)\pi_\bullet(K_s), for each ss.

By exactness an element κ sπ (K s)\kappa_s \in \pi_\bullet(K_s) is in this image if its image

ρ s+1(κ s)π 1(X s+1) \rho_{s+1} \coloneqq \partial(\kappa_s) \in \pi_{\bullet-1}(X_{s+1})

vanishes. Now, by construction of the resolution, “evidence” for this is that f s+1((κ s))π 1(K s+1)f_{s+1}(\partial(\kappa_s)) \in \pi_{\bullet-1}(K_{s+1}) vanishes, which in turn by exactness means equivalently that (κ s)\partial(\kappa_s) is the image of an element ρ s+2π 1(X s+2)π 1(X s+1)\rho_{s+2} \in \pi_{\bullet-1}(X_{s+2}) \to \pi_{\bullet-1}(X_{s+1}). Now again “evidence” for ρ s+2\rho_{s+2} to vanish is that its image f s+2(ρ(s+2))f_{s+2}(\rho(s+2)) vanishes, which again means that it comes from an element ρ s+3π 1(X s+3)π 1(X s+2)\rho_{s+3} \in \pi_{\bullet-1}(X_{s+3}) \to \pi_{\bullet-1}(X_{s+2}).

Proceeding by induction this way, we find that accumulated “evidence” in homotopy groups of K K_\bullet for an element κ s\kappa_s to represent an element in π (X)\pi_\bullet(X) is that its differential κ s\partial \kappa_s factors through all the π 1(X s+k)π 1(X s)\pi_{\bullet-1}(X_{s+k}) \to \pi_{\bullet-1}(X_s). This in turn means that it factors through the inverse limit lim sπ 1(X s)\underset{\leftarrow}{\lim}_s \pi_{\bullet-1}(X_s). Such an element κ s\kappa_s with

κ slim sπ 1(X s)π 1(X s+1) \partial \kappa_s \in \underset{\leftarrow}{\lim}_s \pi_{\bullet-1}(X_s) \to \pi_{\bullet-1}(X_{s+1})

is called a permanent cycle.

In good cases, the Adams resolution is indeed a resolution which means that the inverse limit lim sX s\underset{\leftarrow}{\lim}_s X_s is in fact contractible. This means that all the “evidence” accumulated in a permanent cycle is indeed sufficient evidence to prove the existence of an element σ sπ (X s)\sigma_s \in \pi_\bullet(X_s) and hence of an element σπ (X)\sigma \in \pi_\bullet(X).

A trivial way for this to be the case is that the original σ s\sigma_s is itself in the image under \partial of some element, in which case κ s=0\kappa_s = 0 already all by itself. These elements are called eventual boundaries. Therefore if the Adams resolution is indeed a resolution the quotient group

permanentcycleseventualbounaries \frac{permanent\;cycles}{eventual\;bounaries}

gives elements in π (S)\pi_\bullet(S), and this quotient is what the Adams spectral sequence computes.

Traditional definition

under construction

Given a connective spectrum XX such that H (X)H^\bullet(X) has finite type, then for each prime number pp there exists a spectral sequence which converges to the homotopy groups of XX modulo pp π ast(X) p\pi_ast(X) \otimes \mathbb{Z}_p and whose E 2E^2-page is

E 2 s,tExt A s,t(H (X),/(p)) E_2^{s,t} \simeq Ext_A^{s,t}(H^\bullet(X), \mathbb{Z}/(p) )

where AA is the Steenrod algebra.

This is built via an Adams resolution

X=X 0 g 0 X 1 g 1 X 2 g 2 X 3 f 0 f 1 f 2 f 3 K 0 K 1 K 2 K 2 \array{ X = X_0 &\stackrel{g_0}{\leftarrow}& X_1 &\stackrel{g_1}{\leftarrow}& X_2 &\stackrel{g_2}{\leftarrow}& X_3 &\stackrel{}{\leftarrow}& \cdots \\ \downarrow^{\mathrlap{f_0}} && \downarrow^{\mathrlap{f_1}} && \downarrow^{\mathrlap{f_2}} && \downarrow^{\mathrlap{f_3}} && \\ K_0 && K_1 && K_2 && K_2 }

The long exact sequence induced by this give an exact couple and the Adams spectral sequence is the corresponding spectral sequence.

This is due to (Adams 58). A review is around (Ravenel, theorem, 2.1.1, def. 2.1.8).

Definition in higher algebra

We discuss the general definition of EE-Adams-Novikov spectral sequences for suitable E-∞ rings EE expressed in higher algebra, as in (Lurie, Higher Algebra). 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

Finally we consider specifically the examples of such given by

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 computing homotopy groups of filtered objects

Let thoughout 𝒞\mathcal{C} be a stable (∞,1)-category equipped with a t-structure such that its heart is an abelian category.

Example

For instance

Definition

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

X n+1X nX n1. \cdots X_{n+1} \to X_n \to X_{n-1} \to \cdots \,.

Or rather, the object being filtered is the homotopy limit

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

and the sequential diagram exhibits the filtering.

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

Definition

For a generalized filtered object X X_\bullet, def. 1, write

F nfib(X nX n+1) F_n \coloneqq fib(X_n \to X_{n+1})

for the homotopy fiber of the nnth structure map, for all nn \in \mathbb{Z}, and define an exact couple

π (F ) π (X ) π (X ) \array{ && \pi_\bullet(F_\bullet) \\ & \swarrow && \nwarrow \\ \pi_\bullet(X_\bullet) && \stackrel{}{\longrightarrow} && \pi_\bullet(X_\bullet) }

where the maps are given by the long exact sequences of homotopy groups

π (X n+1)π (F n)π (X n)π (X n+1)π +1(F n) \cdots \to \pi_\bullet(X_{n+1}) \to \pi_\bullet(F_n) \to \pi_\bullet(X_n) \to \pi_\bullet(X_{n+1}) \to \pi_{\bullet+1}(F_n) \to \cdots

We now have the spectral sequence of a filtered stable homotopy type.

Proposition

Let 𝒞\mathcal{C} be a stable (∞,1)-category equipped with a t-structure such that its heart is an abelian category.

If 𝒞\mathcal{C} has sequential limits and if X n0X_n \simeq 0 for all n>n 0n \gt n_0 then the spectral sequence induced by the exact couple of def. 2 converges to the homotopy groups of the homotopy limit lim nX n\underset{\leftarrow}{\lim}_n X_n of the generalized filted object:

E 1 p,q=π p+qF p1π p+q(limX ) E^{p,q}_1 = \pi_{p+q} F_{p-1} \Rightarrow \pi_{p+q} (\underset{\leftarrow}{\lim} X_\bullet)

This is due to (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.

Homotopy groups of cosimplicial totalizations filtered by coskeleta

Definition

Given an cosimplicial object

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

its totalization TotYlim nY nTot Y \simeq \underset{\leftarrow}{\lim}_n Y_n is filtered, def. 1, 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 \,.
Proposition

The filtration spectral sequence, prop. 1, applied to the filtration of a totalization by coskeleta as in def. 3, has as E 2E_2-term the cohomology groups of the Moore complex associated with the cosimplicial object

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) \,.

This is (Higher Algebra, remark 1.2.4.4). Review is around (Wilson 13, theorem 1.2.4).

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 (“Sweedler corings”).

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

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-Sweedler coring”) 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) \,.
Remark

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

Example

The condition in prop. 3 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 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.

We consider now condition 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. 5 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 3 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).

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. 4. Then by prop. 1 and prop. 4 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. 2 and remark 2, the E 2E_2-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) \,.

Properties

Relation to Steenrod algebra

The E 2E_2-page of the Adams spectral sequence for the pp-component of π n+k(S n)\pi_{n+k}(S^n) is the ordinary Steenrod algebra (for given prime pp).

More generally, For RR an E-infinity ring such that its dual RR-Steenrod algebra in the form of the self-homology R (R)R_\bullet(R) is a Hopf algebroid over R =π (R)R_\bullet = \pi_\bullet(R) (see at Steenrod algebra – Hopf algebroid structure), then the E 2E^2-term of the EE-Adams spectral sequence is an Ext of E (E)E_\bullet(E)-comodules

E 2Ext R (R)(R ,R (X)). E^2 \simeq Ext_{R_\bullet(R)}(R_\bullet, R_\bullet(X)) \,.

See the references below.

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

References

General

The original articles are

  • John Adams, On the structure and applications of the Steenrod algebra, Comm. Math. Helv. 32 (1958), 180–214.
  • Sergei Novikov, The methods of algebraic topology from the viewpoint of cobordism theories, Izv. Akad. Nauk. SSSR. Ser. Mat. 31 (1967), 855–951 (Russian).

also

  • Aldridge Bousfield, The localization of spectra with respect to homology , Topology vol 18 (1979) (pdf)

Convergence is nicely treated at the end of

  • A. K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), no. 4, 257–281.

A unique homological perspective is provided in

  • Haynes Miller, On relations between Adams spectral sequences, with an application to the stable homotopy of a Moore space, J. Pure Appl. Algebra 20 (1981)

Trasditional reviews include

also Section A.6 of Ravenel’s orange book 1992

  • Paul Goerss, The Adams-Novikov spectral sequence and the Homotopy Groups of Spheres, lecture notes 2007 (pdf)

See also

  • Nerses Aramian, The Adams spectral sequence (pdf)

  • Alexander Kupers, An introduction to the Adams spectral sequence (pdf)

  • R. Bruner, An Adams spectral sequence primer (pdf)

  • Michael Adamaszek, An elementary guide to the Adams-Novikov ExtExt (pdf)

  • pdf

A nice review from the modern point of view of higher algebra is in

More review along these lines is in

based on

Hopf algebroid ExtExt-structure on E 2E^2

  • Andrew Baker and Alain Jeanneret, Brave new Hopf algebroids and extensions of MUMU-algebras, Homology Homotopy Appl. Volume 4, Number 1 (2002), 163-173. (Euclid)

Revised on January 13, 2014 15:17:14 by Urs Schreiber (89.204.155.62)