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classical Adams spectral sequence

Contents

This entry is about the classical Adams spectral sequence only. For more general discussion see at Adams spectral sequence.

under construction

Contents

Idea

The classical Adams spectral sequence (Adams 58) is a type of spectral sequences used for computations in stable homotopy theory. It computes the homotopy groups of spheres at prime 2 from homology/cohomology, as modules/comodules over its Steenrod operations. The Adams spectral sequence may be seen as a variant of the Serre spectral sequence obtained by replacing a single fibration by an “Adams resolution”.

The original clasical Adams spectral sequenc 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 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 in unstable homotopy theory.

Via iterated Hurewicz theorem and Serre spectral sequence

The classical 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 whole 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 n-sphere S nS^n. For knk \leq n this is clear: first for k<nk \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 beyond 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 spectrum of finite type 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) I:XK 0 iIΣ n iH𝔽 2 f_0 \coloneqq (c_i)_I \;\colon\; X \longrightarrow K_0 \coloneqq \bigvee_{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, then the quotient group

permanentcycleseventualboundaries \frac{permanent\;cycles}{eventual\;boundaries}

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

Properties

Generators in low range

For the moment see at May spectral sequence.

Further

h jh_j is a permanent cycle in the Adams spectral sequence if (2 j)\mathbb{R}^{(2^j)} admits the structure of a real division algebra

Adams: …

d 2(h j+1)=h 0h j 2 d_2(h_{j+1}) = h_0 h_j^2

Mahowald-Tangora: h 4 2h_4^2 is a permanent cycle

Barratt-Jones-Mahowald: h 5 2h_5^2 is a permanent cycle

Hill-Hopkins-Ravenel: for j>7j \gt 7 then h j 2h_j^2 is not a permanent cycle.

References

The original article is

  • Frank Adams, On the structure and applications of the Steenrod algebra, Comm. Math. Helv. 32 (1958), 180–214.

Textbook accounts proceeding in the coalgebra picture include

Further review includes

and

Last revised on May 23, 2016 at 15:33:52. See the history of this page for a list of all contributions to it.