16

CHAPTER 2 Setting up the Adams Spectral Sequence

In this chapter we introduce the spectral sequence that will be our main object of study. We do not intend to give a definitive account of the underlying theory, but merely to make the rest of the book intelligible. Nearly all of this material is due to Adams. The classical Adams spectral sequence [i.e., the one based on ordinary mod (p) cohomology] was first introduced in Adams [3] and a most enjoyable exposition of it can be found in Adams [7]. In Section 1 we give a fairly self-contained account of it, referring to Adams [4] only for standard facts about Moore spectra and inverse limits. We include a detailed discussion of how one extracts differentials from an exact couple and a proof of convergence.

In Section 2 we describe the Adams spectral sequence based on a generalized homology theory E_{∗} satisfying certain assumptions (2.2.5). We rely heavily on Adams [4], referring to it for the more difficult proofs. The E_{∗}-Adams resolutions (2.2.1) and spectral sequences (2.2.4) are defined, the E_{2}-term is identified, and the convergence question is settled (2.2.3). We do not give the spectral sequence in its full generality; we are only concerned with computing π_{∗}(Y), not [X, Y] for spectra X and Y. Most of the relevant algebraic theory, i.e., the study of Hopf algebroids, is developed in Appendix 1.

In Section 3 we study the pairing of Adams spectral sequences induced by a map α: X′ ∧ X′′ → X and the connecting homomorphism associated with a cofibration realizing a short exact sequence in E-homology. Our smash product result implies that for a ring spectrum the Adams spectral sequence is one of differential algebras. To our knowledge these are the first published proofs of these results in such generality.

Throughout this chapter and the rest of the book we assume a working knowledge of spectra and the stable homotopy category as described, for example, in the first few sections of Adams [4].

1. The Classical Adams Spectral Sequence

In this section we will set up the Adams spectral sequence based on ordinary mod (p) cohomology for the homotopy groups of a spectrum X. Unless otherwise stated all homology and cohomology groups will have coefficients in Z/(p) for a prime number p, and X will be a connective spectrum such that H^{∗}(X) (but not necessarily X itself) has finite type.

Recall that H^{∗}(X) is a module over the mod (p) Steenrod algebra A, to be described explicitly in the next chapter. Our object is to prove

2.1.1. Theorem (Adams [3]). Let X be a spectrum as above. There is a spectral sequence

E_{∗}^{∗∗}(X) with d_{r}: E_{r}^{s,t} → E_{r}^{s+r,t+r−1}

such that (a) E_{2}^{s,t} = Ext_{A}^{s,t}(H^{∗}(X), Z/(p)).

(b) if X is of finite type, E_{∞}^{∗∗} is the bigraded group associated with a certain filtration of π_{∗}(X) ⊗ Z_{p}, where Z_{p} denotes the ring of p-adic integers. □

Let E = HZ/(p), the mod (p) Eilenberg–Mac Lane spectrum. We recall some of its elementary properties.

2.1.2. Proposition.

(a) H_{∗}(X) = π_{∗}(E ∧ X).

(b) H^{∗}(X) = [X, E].

(c) H^{∗}(E) = A.

(d) If K is a locally finite wedge of suspensions of E, i.e., a generalized mod (p) Eilenberg–Mac Lane spectrum, then π_{∗}(K) is a graded Z/(p)-vector space with one generator for each wedge summand of K. More precisely, π_{∗}(K) = Hom_{A}(H^{∗}(K), Z/(p)).

(e) A map from X to K is equivalent to a locally finite collection of elements in H^{∗}(X) in the appropriate dimensions. Conversely, any locally finite collection of elements in H^{∗}(X) determines a map to such a K.

(f) If a locally finite collection of elements in H^{∗}(X) generate it as an A-module, then the corresponding map f: X → K induces a surjection in cohomology.

(g) E ∧ X is a wedge of suspensions of E with one wedge summand for each Z/(p) generator of H^{∗}(X). H^{∗}(E ∧ X) = A ⊗ H^{∗}(X) and the map f: X → E ∧ X (obtained by smashing X with the map S^{0} → E) induces the A-module structure map A⊗H^{∗}(X) → H^{∗}(X) in cohomology. In particular H^{∗}(F) is a surjection. □

The idea behind the Adams spectral sequence is to use maps such as those of (f) or (g) and our knowledge of π_{∗}(K) or π_{∗}(E∧X) to get information about π_{∗}(X). We enlist the aid of homological algebra to make the necessary calculations.

More specifically, we have

2.1.3. Definition. A mod (p) Adams resolution (X_{s}, g_{s}) for X is a diagram

X = \xymatrix{X_{0} \ar[d]^{f_{0}} & X_{1} \ar[d]^{f_{1}} \ar[l]_{g_{0}} & X_{2} \ar[d]^{f_{2}} \ar[l]_{g_{1}} & X_{3} \ar[l]_{g_{2}} \\ K_{0} & K_{1} & K_{2}}

where each K_{s}, is a wedge of suspensions of E, H^{∗}(f_{s}) is onto and X_{s+1} is the fiber of f_{s}. □

Proposition 2.1.2(f) and (g) enable us to construct such resolutions for any X, e.g., by setting K_{s} = E ∧ X_{s}. Since H^{∗}(f_{s}) is onto we have short exact sequences

0 ← H^{∗}(X_{s}) ← H^{∗}(K_{s}) ← H^{∗}(ΣX_{s+1}) ← 0.

We can splice these together to obtain a long exact sequence

(2.1.4)  0 ← H^{∗}(X) ← H^{∗}(K_{0}) ← H^{∗}(ΣK_{1}) ← H^{∗}(Σ^{2}K_{2}) ← · · · .

Since the maps are A-module homomorphisms and each H^{∗}(K_{s}) is free over A_{p}, 2.1.4 is a free A-resolution of H^{∗}(X).

Unfortunately, the relation of π_{∗}(K_{s}) to π_{∗}(X) is not as simple as that between the corresponding cohomology groups. Life would be very simple if we knew π_{∗}(f_{s}) was onto, but in general it is not. We have instead long exact sequences

(2.1.5)  [figure: π_{∗}(X_{s+1}); π_{∗}(g_{s}); π_{∗}(X_{s}); π_{∗}(f_{s}); π_{∗}(K_{s}); ∂_{s,∗}]

arising from the fibrations

X_{s+1} ⟶^{g_{s}} X_{s} ⟶^{f_{s}} K_{s}.

If we regard π_{∗}(X_{s}) and π_{∗}(K_{s}) for all s as bigraded abelian groups D_{1} and E_{1}, respectively [i.e., D_{1}^{s,t} = π_{t−s}(X_{s}) and E_{1}^{s,t} = π_{t−s}(K_{s})] then 2.1.5 becomes

(2.1.6)  \xymatrix{D_{1} \ar[rr]^{i_{1}} &  & D_{1} \ar[dl]^{j_{1}} \\  & E_{1} \ar[ul]^{k_{1}}}

where

i_{1} = π_{t−s}(g_{s}): D_{1}^{s+1,t+1} → D_{1}^{s,t},
j_{1} = π_{t−s}(f_{s}): D_{1}^{s,t} → E_{1}^{s,t},

and

k_{1} = ∂_{s,t−s}: E_{1}^{s,t} → D_{1}^{s+1,t}.

The exactness of 2.1.5 translates to ker i_{1} = im k_{1}, ker j_{1} = im i_{1}, and ker k_{1} = im j_{1}. A diagram such as 2.1.6 is known as an exact couple. It is standard homological algebra that an exact couple leads one to a spectral sequence; accounts of this theory can be found in Cartan and Eilenberg [1, Section XV.7], Mac Lane [1, Section XI.5], and Hilton and Stammbach [1, Chapter 8] as well as Massey [2].

Briefly, d_{1} = j_{1}k_{1} : E_{1}^{s,t} → E_{1}^{s+1,t} has (d_{1})^{2} = j_{1}k_{1}j_{1}k_{1} = 0 so (E_{1}, d_{1}) is a complex and we define E_{2} = H(E_{1}, d_{1}). We get another exact couple, called the derived couple,

(2.1.7)  \xymatrix{D_{2} \ar[rr]^{i_{2}} &  & D_{2} \ar[dl]^{j_{2}} \\  & E_{2} \ar[ul]^{k_{2}}}

where D_{2}^{s,t} = i_{1}D_{1}^{s,t}, i_{2} is induced by i_{1}, j_{2}(i_{1}d) = j_{1}d for d ∈ D_{1}, and k_{2}(e) = k_{1}(e) for e ∈ ker d, ⊂ E_{1}. Since 2.1.7 is also an exact couple (this is provable by a diagram chase), we can take its derived couple, and iterating the procedure gives a sequence of exact couples

\xymatrix{D_{r} \ar[rr]^{i_{r}} &  & D_{r} \ar[dl]^{j_{r}} \\  & E_{r} \ar[ul]^{k_{r}}}

where D_{r+1} = i_{r}D_{r}, d_{r} = j_{r}k_{r}, and E_{r+1} = H(E_{r}, d_{r}). The sequences of complexes {(E_{r}, d_{r})} constitutes a spectral sequence. A close examination of the indices will reveal that d_{r}: E_{r}^{s,t} → E_{r}^{s+r,t+r−1}. It follows that for s < r, the image of d_{r} in E_{r}^{s,t} is trivial so E_{r+1}^{s,t} is a subgroup of E_{r}^{s,t}, hence we can define

E_{∞}^{s,t} = ⋂_{r>s} E_{r}^{s,t}.

This group will be identified (2.1.12) in certain cases with a subquotient of π_{t−s}(X), namely, im π_{t−s}(X_{s})/ im π_{t−s}(X_{s+1}). The subgroups im π_{∗}(X_{s}) = F^{s}π_{∗}(X) form a decreasing filtration of π_{∗}(X) and E_{∞} is the associated bigraded group.

2.1.8. Definition. The mod (p) Adams spectral sequence for X is the spectral sequence associated to the exact couple 2.1.6. □

We will verify that d_{r}: E_{r}^{s,t} → E_{r}^{s+r,t+r−1} by chasing diagram 2.1.9, where we write π_{∗}(X_{∗}) and π_{∗}(K_{∗}) instead of D_{1} and E_{1}, with u = t − s. (2.1.9)

[figure: π_{u}(X_{s+2}); π_{u}(f_{s+2}); π_{u}(g_{s+1}); π_{u}(K_{s+2}); ∂_{s+2,u}; π_{u−1}(X_{s+3}); π_{u−1}(f_{s+3}); π_{u−1}(g_{s+2}); π_{u−1}(K_{s+3}); π_{u}(X_{s+1}); π_{u}(f_{s+1}); π_{u}(g_{s}); π_{u}(K_{s+1}); ∂_{s+1,u}; π_{u−1}(X_{s+2}); π_{u−1}(f_{s+2}); π_{u−1}(g_{s+1}); π_{u−1}(K_{s+2}); π_{u}(X_{s}); π_{u}(f_{s}); π_{u}(K_{s}); ∂_{s,u}; π_{u−1}(X_{s+1}); π_{u−1}(f_{s+1}); π_{u−1}(K_{s+1})]

The long exact sequences 2.1.5 are embedded in this diagram; each consists of a vertical step π_{∗}(g_{∗}) followed by horizontal steps π_{∗}(f_{∗}) and ∂_{∗∗} and so on. We have E_{1}^{s,t} = π_{u}(K_{s}) and d_{1}^{s,t} = (π_{u−1}(f_{s+1}))(∂_{s,u}). We have E_{2}^{s,t} = ker d_{1}^{s,t}/ im d_{1}^{s−1,t}. Suppose an element in E_{2}^{s,t} is represented by x ∈ π_{u}(K_{s}). We will now explain how d_{2}[x] (where [x] is the class represented by x) is defined. x is a d_{1} cycle, i.e., d_{1}x = 0, so exactness in 2.1.4 implies that ∂_{s,u}x = (π_{u+1}(g_{s+1}))(y) for some y ∈ π_{u−1}(X_{s+2}). Then (π_{u−1}(f_{s+2}))(y) is a d_{1} cycle which represents d_{2}[x] ∈ E_{2}^{s+2,t−1}. If d_{2}[x] = 0 then [x] represents an element in E_{3}^{s,t} which we also denote by [x]. To define d_{3}[x] it can be shown that y can be chosen so that y = (π_{u−1}(g_{s+2}))(y′) for some y′ ∈ π_{u−1}(X_{s+3}) and that (π_{u−1}(f_{s+3}))(y′) is a d_{1} cycle representing a d_{2} cycle which represents an element in E^{s+3,t+2} which we define to be d_{3}[x]. These assertions may be verified by drawing another diagram which is related to the derived couple 2.1.7 in the same way that 2.1.9 is related to the original exact couple 2.1.6. The higher differentials are defined in a similar fashion. In practice, even the calculation of d_{2} is a delicate business.

Before identifying E_{∞}^{s,t} we need to define the homotopy inverse limit of spectra.

2.1.10. Definition. Given a sequence of spectra and maps

X_{0} ⟵^{f_{1}} X_{1} ⟵^{f_{2}} X_{2} ⟵^{f_{3}} X_{3} ⟵ ⋯,

\underleftarrow{lim} X_{i}, is the fiber of the map

g: ∏ X_{i} → ∏ X_{i}

whose ith component is the difference between the projection p_{i}: ∏ X_{j} → X_{i} and the composite

∏ X_{j} ⟶^{p_{i+1}} X_{i+1} ⟶^{f_{i+1}} X_{i}.  □

For the existence of products in the stable category see 3.13 of Adams [4]. This \underleftarrow{lim} is not a categorical inverse limit (Mac Lane [1, Section III.4] because a compatible collection of maps to the X_{i}, does not give a unique map to \underleftarrow{lim} X_{i}. For this reason some authors (e.g., Bousfield and Kan [1]) denote it instead by \underleftarrow{holim}. The same can be said of the direct limit, which can be defined as the cofiber of the appropriate self-map of the coproduct of the spectra in question. However this \underleftarrow{lim} has most of the properties one would like, such as the following.

2.1.11. Lemma. Given spectra X_{i,j} for i, j ≥ 0 and maps f: X_{i,j} → X_{i−1,j} and g: X_{i,j} → X_{i,j−1} such that fg is homotopic to gf,

\underleftarrow{lim}_{i} \underleftarrow{lim}_{j} X_{i,j} = \underleftarrow{lim}_{j} \underleftarrow{lim}_{i} X_{i,j}.

Proof. We have for each i a cofibre sequence

\underleftarrow{lim}_{j} X_{i,j} → ∏_{j} X_{i,j} → ∏_{j} X_{i,j}.

Next we need to know that products preserve cofiber sequences. For this fact, recall that the product of spectra ∏ Y_{i}, is defined via Brown’s representability theorem (Adams [4], Theorem 3.12) as the spectrum representing the functor ∏[−, Y_{i}]. Hence the statement follows from the fact that a product (although not the inverse limit) of exact sequences is again exact.

Hence we get the following homotopy commutative diagram in which both rows and columns are cofiber sequences.

\xymatrix{\underleftarrow{lim}_{i} \underleftarrow{lim}_{j} X_{i,j} \ar[r] \ar[d] & \underleftarrow{lim}_{i} ∏_{j} X_{i,j} \ar[r] \ar[d] & \underleftarrow{lim}_{i} ∏ X_{i,j} \ar[d] \\ ∏_{i} \underleftarrow{lim}_{j} X_{i,j} \ar[r] \ar[d] & ∏_{i} ∏_{j} X_{i,j} \ar[r] \ar[d] & ∏_{i} ∏_{j} X_{i,j} \ar[d] \\ ∏_{i} \underleftarrow{lim}_{j} X_{i,j} \ar[r] & ∏_{i} ∏_{j} X_{i,j} \ar[r] & ∏_{i} ∏_{j} X_{i,j}}

Everything in sight is determined by the two self-maps of ∏_{i} ∏_{j} X_{i,j} and the homotopy that makes them commute. Since the product is categorical we have ∏_{i} ∏_{j} X_{i,j} = ∏_{j} ∏_{i} X_{i,j}. It follows that ∏_{i} \underleftarrow{lim}_{j} X_{i,j} = \underleftarrow{lim}_{j} ∏_{i} X_{i,j} because they are each the fiber of the same map.

Similarly

∏_{j} \underleftarrow{lim}_{i} X_{i,j} = \underleftarrow{lim}_{i} ∏_{j} X_{i,j}

so one gets an equivalent diagram with \underleftarrow{lim}_{j} \underleftarrow{lim}_{i} X_{i,j} in the upper left corner. □

Now we will show that for suitable X, E_{∞}^{s,t} is a certain subquotient of π_{u}(X).

2.1.12. Lemma. Let X be a spectrum with an Adams resolution (X_{s}, g_{s}) such that \underleftarrow{lim} X_{s} = pt. Then E_{∞}^{s,t} is the subquotient im π_{u}(X_{s})/ im π_{u}(X_{s+1}) of π_{u}(X) and ⋂ im π_{∗}(X_{s}) = 0.

Proof. For the triviality of the intersection we have \underleftarrow{lim} π_{∗}(X_{s}) = 0 since \underleftarrow{lim} X_{s} = pt. Let G_{s} = π_{∗}(X_{s}) and

G_{s}^{t} = \begin{cases}G_{s} & if s ≥ t \\ im G_{t} < G_{s} & if t ≥ s. \end{cases}

We have injections G_{s}^{t} → G_{s}^{t−1} and surjections G_{s}^{t} → G_{s−1}^{t}, so \underleftarrow{lim}_{t} G_{s}^{t} = ⋂_{t} G_{s}^{t} and \underleftarrow{lim}_{s} G_{s}^{t} = G_{t}. We are trying to show \underleftarrow{lim}_{t} G_{0}^{t} = 0. \underleftarrow{lim}_{t} G_{s}^{t} maps onto \underleftarrow{lim}_{t} G_{s−1}^{t}, so \underleftarrow{lim}_{s} \underleftarrow{lim}_{t} G_{s}^{t} maps onto \underleftarrow{lim}_{t} G_{0}^{t}. But \underleftarrow{lim}_{s} \underleftarrow{lim}_{t} G_{s}^{t} = \underleftarrow{lim}_{t} \underleftarrow{lim}_{s} G_{s}^{t} = \underleftarrow{lim}_{t} G_{t} = 0.

For the identification of E_{∞}^{s,t}, let 0 ≠ [x] ∈ E_{∞}^{s,t}.

First we show ∂_{s,u}(x) = 0. Since d_{r}[x] = 0, ∂_{s,u}(x) can be lifted to π_{u−1}(X_{s+r+1}) for each r. It follows that ∂_{s,u}(x) ∈ im \underleftarrow{lim} π_{u−1}(X_{s+r}) = 0, so ∂_{s,u}(x) = 0.

Hence we have x = π_{u}(f_{s})(y) for y ∈ π_{u}(X_{s}). It suffices to show that y has a nontrivial image in π_{u}(X). If not, let r be the largest integer such that y has a nontrivial image z ∈ π_{u}(X_{s−r+1}). Then z = ∂_{s−r,u}(w) for w ∈ π_{u}(K_{s−r}) and d_{r}[w] = [x], contradicting the nontriviality of [x]. □

Now we prove 2.1.1(a), the identification of the E_{2}-term.

By 2.1.2(d), E_{1}^{s,t} = Hom_{A}(H^{t−s}(K_{s}), Z/(p)). Hence applying Hom_{A}(−, Z/(p)) to 2.1.4 gives a complex

E_{1}^{0,t} ⟶^{δ} E_{1}^{1,t} ⟶^{δ} E_{1}^{2,t} → · · · .

The cohomology of this complex is by definition the indicated Ext group. It is straightforward to identify the coboundary δ with the d_{1} in the spectral sequence and 2.1.1(a) follows.

2.1.13. Corollary. If f: X → Y induces an isomorphism in mod (p) homology then it induces an isomorphism (from E_{2} onward) in the mod (p) Adams spectral sequence. □

2.1.14. Definition. Let G be an abelian group and X a spectrum. Then XG = X ∧ SG, where SG is the Moore spectrum associated with G (Adams [4, p. 200]. Let X̂ = XZ_{p} (the p-aidc completion of X), where Z_{p} is the p-adic integers, and X^{m} = XZ/(p^{m}). □

2.1.15. Lemma. (a) The map X → X̂ induces an isomorphism of mod (p) Adams spectral sequences.

(b) π_{∗}(X̂) = π_{∗}(X) ⊗ Z_{p}.

(c) X̂ = \underleftarrow{lim} X^{m}, if x has finite type.

Proof. For (a) it suffices by 2.1.11 to show that the map induces an isomorphism in mod (p) homology. For this see Adams [4], proposition 6.7, which also shows (b).

Part (c) does not follow immediately from the fact that SZ_{p} = \underleftarrow{lim} SZ/(p^{m}) because inverse limits do not in general commute with smash products. Indeed our assertion would be false for X = SQ, but we are assuming that X has finite type.

By 2.1.10 there is a cofibration

SZ_{p} → ∏_{m} SZ/(p^{m}) → ∏_{m} SZ/(p^{m}),

so it suffices to show that

X ∧ ∏ SZ/(p^{m}) ≃ ∏ XZ/(p^{m}).

This is a special case (with X = E and R = Z) of Theorem 15.2 of Adams [4]. □

2.1.16. Lemma. If X is a connective spectrum with each π_{i}(X) a finite p-group, then for any mod (p) Adams resolution (X_{s}, g_{s}) of X, \underleftarrow{lim} X_{s} = pt.

Proof. Construct a diagram

X = X′_{0} ← X′_{1} ← X′_{2} ← ⋯

(not an Adams resolution) by letting X′_{s+1} be the fiber in

X′_{s+1} → X′_{s} → K_{s},

where the right-hand map corresponds [2.1.2(e)] to a basis for the bottom cohomology group of X_{s}. Then the finiteness of π_{i}(X) implies that for each i, π_{i}(X′_{s}) = 0 for large s. Moreover, π_{∗}(X′_{s+1}) → π_{∗}(X′_{s}) is monomorphic so \underleftarrow{lim} X′_{s} = pt.

Now if (X_{s}, g_{s}) is an Adams resolution, the triviality of g_{s} in cohomology enables us to construct compatible maps X_{s} → X′_{s}. It follows that the map \underleftarrow{lim} π_{∗}(X_{s}) → π_{∗}(X) is trivial. Each X_{s} also satisfies the hypotheses of the lemma, so we conclude that \underleftarrow{lim} π_{∗}(X_{s}) has trivial image in each π_{∗}(X_{s}) and is therefore trivial. Since π_{i}(X_{s}) is finite for all i and s, \underleftarrow{lim}^{1} π_{∗}(X_{s}) = 0 so \underleftarrow{lim} X_{s} = pt. □

We are now ready to prove 2.1.1(b), i.e., to identify the E_{∞}-term. By 2.1.15(a) it suffices to replace X by X̂. Note that since SZ_{p} ∧ SZ/(p^{m}) = SZ/(p^{m}), X^{m} = X̂^{m}. It follows that given a mod (p) Adams resolution (X_{s}, g_{s}) for X, smashing with SZ_{p} and SZ/(p^{m}) gives resolutions (X̂_{s}, ĝ_{s}) and (X_{s}^{m}, g_{s}^{m}) for X̂ and X^{m}, respectively. Moreover, X^{m} satisfies 2.1.16 so \underleftarrow{lim}_{s} X_{s}^{m} = pt. Applying 2.1.15(c) to each X_{s}, we get X̂_{s} = \underleftarrow{lim}_{m} X_{s}^{m}, so

\underleftarrow{lim}_{s} X_{s} = \underleftarrow{lim}_{s} \underleftarrow{lim}_{m} X_{s}^{m}
= \underleftarrow{lim}_{m} \underleftarrow{lim}_{s} X_{s}^{m} by 2.1.11
= pt.

Hence the result follows from 2.1.12. □

2.1.17. Remark. The E_{∞} term only gives us a series of subquotients of π_{∗}(X) ⊗ Z_{p}, not the group itself. After computing E_{∞} one may have to use other methods to solve the extension problem and recover the group.

We close this section with some examples.

2.1.18. Example. Let X = HZ, the integral Eilenberg–Mac Lane spectrum. The fundamental cohomology class gives a map f: X → E with H^{∗}(f) surjective. The fiber of f is also X, the inclusion map g: X → X having degree p. Hence we get an Adams resolution (2.1.3) with X_{s} = X and K_{s} = E for all s, the map X = X_{s} → X_{0} = X, having degree p^{s}. We have then

E_{1}^{s,t} = \begin{cases}Z/(p) & if t = s \\ 0 & if t ≠ s. \end{cases}

There is no room for nontrivial differentials so the spectral sequence collapses, i.e., E_{∞} = E_{1}. We have E_{∞}^{s,s} = Z/(p) = im π_{0}(X_{s})/ im π_{0}(X_{s+1}). In this case X̂ = HZ_{p}, the Eilenberg–Mac Lane spectrum for Z_{p}.

2.1.19. Example. Let X = HZ/(p^{i}) with i > 1. It is known that H^{∗}(X) = H^{∗}(Y)⊕ΣH^{∗}(Y) as A-modules, where Y = HZ. This splitting arises from the two right-hand maps in the cofiber sequence

Y → Y → X → ΣY,

where the left-hand map has degree p^{i}. Since the E_{2}-term of the Adams spectral sequence depends only on H^{∗}(X) as an A-module, the former will enjoy a similar splitting. In the previous example we effectively showed that

Ext_{A}^{s,t}(H^{∗}(Y), Z/(p)) = \begin{cases}Z/(p) & if t = s \\ 0 & if t ≠ s. \end{cases}

It follows that in the spectral sequence for X we have

E_{2}^{s,t} = \begin{cases}Z/(p) & if t − s = 0 or 1 \\ 0 & otherwise \end{cases}

In order to give the correct answer we must have E_{∞}^{s,t} = 0 if t − s = 1 and E_{∞}^{s,t} = 0 if t = s for all but i values of s. Multiplicative properties of the spectral sequence to be discussed in Section 3 imply that the only way we can arrive at a suitable E_{∞} term is to have d_{i}: E_{i}^{s,s+1} → E_{i}^{s+i,s+i} nontrivial for all s ≥ 0. A similar conclusion can be drawn by chasing the relevant diagrams.

2.1.20. Example. Let X be the fiber in X → Ŝ^{0} → HZ_{p} where the right-hand map is the fundamental integral cohomology class on S^{0}. Smashing the above fibration with X we get

X ∧ X ⟶^{g_{0}} X ⟶^{f_{0}} X ∧ HZ

It is known that the integral homology of X has exponent p, so X ∧ HZ is a wedge of E and H^{∗}(f_{0}) is surjective. Similar statements hold after smashing with X any number of times, so we get an Adams resolution (2.1.3) with K_{s} = X_{s} ∧ HZ and X_{s} = X^{(s+1)}, the (s + 1)-fold smash product of X with itself, i.e., one of the form

\xymatrix{X \ar[d] & X ∧ X \ar[l] \ar[d] & X ∧ X ∧ X \ar[l] & ⋯ \ar[l] \\ X ∧ HZ & X ∧ X ∧ HZ.}

Since X is (2p − 4)-connected X_{s}, is ((s + 1)(2p − 3) − 1)-connected, so \underleftarrow{lim} X_{s}, is contractible.

2. The Adams Spectral Sequence Based on a Generalized Homology Theory

In this section we will define a spectral sequence similar to that of 2.1.1 (the classical Adams spectral sequence) in which the mod (p) Eilenberg–Mac Lane spectrum is replaced by some more general spectrum E. The main example we have in mind is of course E = BP, the Brown–Peterson spectrum, to be defined in 4.1.12. The basic reference for this material is Adams [4] (especially Section 15, which includes the requisite preliminaries on the stable homotopy category.

Our spectral sequence should have the two essential properties of the classical one: it converges to π_{∗}(X) localized or completed at p and its E_{2}-term is a functor of E^{∗}(X) (the generalized cohomology of X) as a module over the algebra of cohomology operations E^{∗}(E); i.e., the E_{2}-term should be computable in some homological way, as in 2.1.1. Experience has shown that with regard to the second property we should dualize and consider instead E_{∗}(X) (the generalized homology of X) as a comodule over E_{∗}(E) (sometimes referred to as the coalgebra of cooperations). In the classical case, i.e., when E = HZ/(p), E_{∗}(E) is the dual Steenrod algebra A_{∗}.

Theorem 2.1.1(a) can be reformulated as E_{2} = Ext_{A_{∗}}(Z/(p), H_{∗}(X)) using the definition of Ext in the category of comodules given in A1.2.3. In the case E = BP substantial technical problems can be avoided by using homology instead of cohomology. Further discussion of this point can be found in Adams [6, pp. 51–55].

Let us assume for the moment that we have known enough about E and E_{∗}(E) to say that E_{∗}(X) is a comodule over E_{∗}(E) and we have a suitable definition of Ext_{E_{∗}(E)}(E_{∗}(S^{0}), E_{∗}(X)), which we abbreviate as Ext(E_{∗}(X)). Then we might proceed as follows.

2.2.1. Definition. An E_{∗}-Adams resolution for X is a diagram

[figure: X = X_{0}; f_{0}; X_{1}; f_{1}; g_{0}; X_{2}; f_{2}; g_{1}; ⋯; K_{0}; K_{1}; K_{2}]

such that for all s ≥ 0 the following conditions hold.

(a) X_{s+1} is the fiber of f_{s}.

(b) E ∧X_{s} is a retract of E ∧K_{s}, i.e., there is a map h_{s}: E ∧K_{s} → E ∧X_{s} such that h_{s}(E ∧f_{s}) is an identity map of E ∧X_{s}. particular E_{∗}(f_{s}) is a monomorphism.

(c) K_{s} is a retract of E ∧ K_{s}.

(d)

Ext^{t,u}(E_{∗}(K_{s})) = \begin{cases}π_{u}(K_{s}) & if t = 0 \\ 0 & if t > 0. \end{cases}  □

As we will see below, conditions (b) and (c) are necessary to insure that the spectral sequence is natural, while (d) is needed to give the desired E_{2}-term. As before it is convenient to consider a spectrum with the following properties.

2.2.2. Definition. An E-completion X̂ of X is a spectrum such that

(a) There is a map X → X̂ inducing an isomorphism in E_{∗}-homology.

(b) X̂ has an E_{∗}-Adams resolution \{X̂_{s}\} with \underleftarrow{lim} X̂_{s} = pt. □

This is not necessarily the same as the X̂ of 2.1.14, which will be denoted in this section by X_{p} (2.2.12). Of course, the existence of such a spectrum (2.2.13) is not obvious and we will not give a proof here. Assuming it, we can state the main result of this section.

2.2.3. Theorem (Adams [4]). An E_{∗}-Adams resolution for X (2.2.1) leads to a natural spectral sequence E_{∗}^{∗∗}(X) with d_{r}: E_{r}^{s,t} → E_{r}^{s+r,t+r−1} such that

(a) E_{2}^{s,t} = Ext(E_{∗}(X)).

(b) E_{∞}^{∗∗} is the bigraded group associated with a certain filtration of π_{∗}(X̂), in other words, the spectral sequence converges tothelatter. (This filtration will be described in 2.2.14.)

2.2.4. Definition. The spectral sequence of 2.2.3 is the Adams spectral sequence for X based on E-homology. □

2.2.5. Assumption. We now list the assumptions on E which will enable us to define Ext and X̂.

(a) E is a commutative associative ring spectrum.

(b) E is connective, i.e., π_{r}(E) = 0 for r < 0.

(c) The map μ_{∗}: π_{0}(E) ⊗ π_{0}(E) → π_{0}(E) induced by the multiplication μ: E ∧ E → E is an isomorphism.

(d) E is flat, i.e., E_{∗}(E) is flat as a left module over π_{∗}(E).

(e) Let θ: Z → π_{0}(E) be the unique ring homomorphism, and let R ⊂ Q be the largest subring to which θ extends. Then H_{r}(E; R) is finitely generated over R for all r.

2.2.6. Proposition. HZ/(p) and BP satisfy 2.2.5(a)–(e) □

The flatness condition 2.2.5(d) is only necessary for identifying E_{2}^{∗∗} as an Ext. Without it one still has a spectral sequence with the specified convergence properties. Some well-known spectra which satisfy the remaining conditions are HZ, bo, bu, and MSU. In these cases E ∧ E is not a wedge of suspensions of E as it is when E = HZ/(p), BP, or MU. HZ ∧ HZ is known to be a certain wedge of suspensions of HZ/(p) and HZ, bo ∧ bo is described by Milgram [1], bu ∧ bu by Adams [4], Section 17, and MSU ∧ MSU by Pengelley [1].

We now turn to the definition of Ext. It follows from our assumptions 2.2.5 that E_{∗}(E) is a ring which is flat as a left π_{∗}(E) module. Moreover, E_{∗}(E) is a π_{∗}(E) bimodule, the right and left module structures being induced by the maps

E = S^{0} ∧ E → E ∧ E and E = E ∧ S^{0} → E ∧ E,

respectively. In the case E = HZ/(p) these two module structures are identical, but not when E = BP. Following Adams [4], Section 12, let μ: E ∧ E be the multiplication on E and consider the map

(E ∧ E) ∧ (E ∧ X) ⟶^{1∧μ∧1} E ∧ E ∧ X.

2.2.7. Lemma. The above map induces an isomorphism

E_{∗}(E) ⊗_{π_{∗}(E)} E_{∗}(X) → π_{∗}(E ∧ E ∧ X).

Proof. The result is trivial for X = S^{n}. It follows for X finite by induction on the number of cells using the 5-lemma, and for arbitrary X by passing to direct limits. □

Now the map

E ∧ X = E ∧ S^{0} ∧ X → E ∧ E ∧ X

induces

ψ: E_{∗}(X) → π_{∗}(E ∧ E ∧ X) = E_{∗}(E) ⊗_{π_{∗}(E)} E_{∗}(X).

In particular, if X = E we get

Δ: E_{∗}(E) → E_{∗}(E) ⊗_{π^{∗}(E)} E_{∗}(E).

Thus E_{∗}(E) is a coalgebra over π_{∗}(E) as well as an algebra, and E_{∗}(X) is a comodule over E_{∗}(E). One would like to say that E_{∗}(E), like the dual Steenrod algebra, is a commutative Hopf algebra, but that would be incorrect since one uses the bimodule structure in the tensor product E_{∗}(E) ⊗_{π_{∗}(E)} E_{∗}(E) (i.e., the product is with respect to the right module structure on the first factor and the left module structure on the second). In addition to the coproduct Δ and algebra structure, it has a right and left unit η_{R}, η_{L}: π_{∗}(E) → E_{∗}(E) corresponding to the two module structures, a counit ε: E_{∗}(E) → π_{∗}(E) induced by μ: E ∧ E → E, and a conjugation c: E_{∗}(E) → E_{∗}(E) induced by interchange the factors in E ∧ E.

2.2.8. Proposition. With the above structure maps (π_{∗}(E), E_{∗}(E)) is a Hopf algebroid (A1.1.1), and E-homology is a functor to the category of left E_{∗}(E)- comodules (A1.1.2), which is abelian (A1.1.3). □

The problem of computing the relevant Ext groups is discussed in Appendix 1, where an explicit complex (the cobar complex A1.2.11) for doing so is given. This complex can be realized geometrically by the canonical E_{∗}-Adams resolution defined below.

2.2.9. Lemma. Let K_{s} = E ∧ X_{s}, and let X_{s+1} be the fiber of f_{s}: X_{s} → K_{s}. Then the resulting diagram (2.2.1) is an E_{∗}-Adams resolution for X.

Proof. Since E is a ring spectrum it is a retract of E∧E, so E∧X_{s}, is a retract of E ∧ K_{s} = E ∧ E ∧ X_{s} and 2.2.1(b) is satisfied. E ∧ X_{s} is an E-module spectrum so 2.2.1(c) is satisfied. For 2.2.1(d) we have E_{∗}(K_{s}) = E_{∗}(E) ⊗_{π_{∗}(E)} E_{∗}(X_{s}) by 2.2.7 and Ext(E_{∗}(K_{s})) has the desired properties by A1.2.1 and A1.2.4. □

2.2.10. Definition. The canonical E_{∗}-Adams resolution for X is the one given by 2.2.9.

Note that if E is not a ring spectrum then the above f_{s} need not induce a monomorphism in E-homology, in which case the above would not be an Adams resolution.

Note also that the canonical resolution for X can be obtained by smashing X with the canonical resolution for S^{0}.

2.2.11. Proposition. The E_{1}-term of the Adams spectral sequence associated with the resolution of 2.2.9 is the cobar complex C^{∗}(E_{∗}(X)) (A1.2.11). □

Next we describe an E-completion X̂ (2.2.2). First we need some more terminology.

2.2.12. Definition. X_{(p)} = XZ_{(p)}, where Z_{(p)} denotes the integers localized at p, and X_{p} = XZ_{p} (see 2.1.14).

2.2.13. Theorem. If X is connective and E satisfies 2.2.5(a)–(e) then an Ecompletion (2.2.2) of X is given by

X̂ = \begin{cases}XQ & if π_{0}(E) = Q \\ X_{(p)} & if π_{0}(E) = Z_{(p)} \\ X & if π_{0}(E) = Z \\ X_{p} & if π_{0}(E) = Z/(p) and π_{n}(X) \\ & is finitely generated for all n. \end{cases}  □

These are not the only possible values of π_{0}(E), but the others will not concern us. A proof is given by Adams [4], Theorem 14.6 and Section 15. We will sketch a proof using the additional hypothesis that π_{1}(E) = 0, which is true in all of the cases we will consider in this book.

For simplicity assume that π_{0}(X) is the first nonzero homotopy group. Then in the cases where π_{0}(E) is a subring of Q we have π_{i}(X̂ ∧ \overline{E}^{(s)}) = 0 for i < s, so by setting X̂_{s} = X̂ ∧ \overline{E}^{(s)} we get \underleftarrow{lim} X̂_{s} = pt.

The remaining case, π_{0}(E) = Z/(p) can be handled by an argument similar to that of the classical case. We show XZ/(p^{m}) is its own E-completion by modifying the proof of 2.1.16 appropriately. Then X_{p} can be shown to be E-complete just as in the proof of 2.1.1(b) (following 2.1.16).

Now we are ready to prove 2.2.3(a). As in Section 1 the diagram 2.2.1 leads to an exact couple which gives the desired spectral sequence. To identify the E_{2}-term, observe that 2.2.1(a) implies that each fibration in the resolution gives a short (as opposed to long) exact sequence in E-homology. These splice together to give a long exact sequence replacing 2.1.3,

0 → E_{∗}(X) → E_{∗}(K_{0}) → E_{∗}(ΣK_{1}) → · · · .

Condition 2.2.1(c) implies that the E_{2}-term of the spectral sequence is the cohomology of the complex

Ext^{0}(E_{∗}(K_{0})) → Ext^{0}(E_{∗}(ΣK_{1})) → · · · .

By A1.2.4 this is Ext(E_{∗}(X)).

For 2.2.3(b) we know that the map X → X̂ induces a spectral sequence isomorphism since it induces an E-homology isomorphism. We also know that \underleftarrow{lim} X̂_{s} = pt, so we can identify E_{∞}^{∗∗} as in 2.1.12.

We still need to show that the spectral sequence is natural and independent (from E_{2} onward) of the choice of resolution. The former implies the latter as the identity map on X induces a map between any two resolutions and standard homological arguments show that such a map induces an isomorphism in E_{2} and hence in E_{r} for r ≥ 2. The canonical resolution is clearly natural so it suffices to show that any other resolution admits maps to and from the canonical one.

We do this in stages as follows. Let {f_{s}: X_{s} → K_{s}} be an arbitrary resolution and let R^{0} be the canonical one. Let R^{n} = {f_{s}^{n}: X_{s}^{n} → K_{s}^{n}} be defined by X_{s}^{n} = X_{s}, and K_{s}^{n} = K_{n} for s < n and K_{s}^{n} = E ∧ X_{s}^{n}; for s ≥ n. Then R^{∞} is the arbitrary resolution and we construct maps R^{0} ↔ R^{∞} by constructing maps R^{n} ↔ R^{n+1}, for which it suffices to construct maps between K_{s} and E ∧ X_{s} compatible with the map from X_{s}. By 2.2.1(b) and (c), K_{s} and E ∧ X_{s} are both retracts of E ∧ K_{s}, so we have a commutative diagram

\xymatrix{X_{s} \ar[r] \ar[d] & K_{s} \ar[d] \ar[dr] \\ E ∧ X_{s} \ar[r] \ar[dr] & E ∧ K_{s} \ar[r] \ar[d] & E ∧ X_{s} \\  & K_{s}}

in which the horizontal and vertical composite maps are identities. It follows that the diagonal maps are the ones we want.

The Adams spectral sequence of 2.2.3 is useful for computing π_{∗}(X), i.e., [S^{0}, X]. With additional assumptions on E one can generalize to a spectral sequence for computing [W, X]. This is done in Adams [4] for the case when E_{∗}(W) is projective over π_{∗}(E). We omit this material as we have no need for it.

Now we describe the filtration of 2.2.3(b), which will be referred to as the E_{∗}-Adams filtration on π_{∗}(X̂).

2.2.14. Filtration Theorem. The filtration on π_{∗}(X̂) of 2.2.3(b) is as follows. A map f: S^{n} → X has filtration ≥ s if f can be factored into s maps each of which becomes trivial after smashing the target with E.

Proof. We have seen above that F^{s}π_{∗}(X̂) = im π_{∗}(X_{s}). We will use the canonical resolution (2.2.10). Let \overline{E} be the fiber of the unit map S^{0} → E. Then X_{2} = \overline{E}^{(s)} ∧ X, where \overline{E}^{(s)} is the s-fold smash product of \overline{E}. X_{i+1} → X_{i} → X_{i} ∧ E is a fiber sequence so each such composition is trivial and a map S^{n} → X which lifts to X_{s} clearly satisfies the stated condition. It remains to show the converse, i.e., that if a map f: S^{n} → X factors as

S^{n} → Y_{s} ⟶^{g_{s}} Y_{s−1} ⟶^{g_{s−1}} ⋯ → Y_{0} = X,

where each composite Y_{i} ⟶^{g_{i}} Y_{i−1} → Y_{i−1} ∧ E is trivial, then it lifts to X_{s}. We argue by induction on i. Suppose Y_{i−1} → X lifts to X_{i−1} (a trivial statement for i = 1). Since Y_{i} maps trivially to Y_{i−1} ∧ E, it does so to X_{i−1} ∧ E and therefore lifts to X_{i}. □

3. The Smash Product Pairing and the Generalized Connecting Homomorphism

In this section we derive two properties of the Adams spectral sequence which will prove usefull in the sequel. The first concerns the structure induced by a map

(2.3.1)  α: X′ ∧ X′′ → X,

e.g., the multiplication on a ring spectrum. The second concerns a generalized connecting homomorphism arising from a cofiber sequence

(2.3.2)  W ⟶^{f} X ⟶^{g} Y ⟶^{h} ΣW

when E_{∗}(h) = 0. Both of these results are folk theorems long known to experts in the field but to our knowledge never before published in full generality. The first property in the classical case was proved in Adams [3], while a weaker form of the second property was proved by Johnson, Miller, Wilson, and Zahler [1].

Throughout this section the assumptions 2.2.5 on E will apply. However, the flatness condition [2.2.5(d)] is only necessary for statements explicitly involving Ext, i.e., 2.3.3(e) and 2.3.4(a). For each spectrum X let E_{∗}^{∗∗}(X) be the Adams spectral sequence for X based on E-homology (2.2.3). Our first result is

2.3.3. Theorem. Let 2 ≤ r ≤ ∞. Then the map a above induces a natural pairing

E_{r}^{∗∗}(X′) ⊗ E_{r}^{∗∗}(X′′) → E_{r}^{∗∗}(X)

such that (a) for a′ ∈ E_{r}^{s′,t′}(X′), a′′ ∈ E_{r}^{s′′,t′′}(X′′),

d_{r}(a′, a′′) = d_{r}(a′)a′′ + (−1)^{t′−s′}a′d_{r}(a′′);

(b) the pairing on E_{r+1}, is induced by that on E_{r};

(c) the pairing on E_{∞}, corresponds to a_{∗}: π_{∗}(X′) ⊗ π_{∗}(X′′) → π_{∗}(X);

(d) if X′ = X′′ = X and E_{∗}(α): E_{∗}(X) ⊗ E_{∗}(X) → E_{∗}(X) is commutative or associative, then so is the pairing [modulo the usual sign conventions, i.e., a′a′′ = (−1)^{(t′−s′)(t′′−s′′)}a′′a′];

(e) for r = 2 the pairing is the external cup product (A1.2.13)

Ext(E_{∗}(X′)) ⊗ Ext(E_{∗}(X′′)) → Ext(E_{∗}(X′) ⊗_{π_{∗}(E)} E_{∗}(X′′))

composed with the map in Ext induced by the composition of canonical maps

E_{∗}(X′) ⊗_{π_{∗}(E)} E_{∗}(X′′) → E_{∗}(X′ ∧ X′′) ⟶^{α_{∗}} E_{∗}(X).

In particular, by setting X′ = S^{0} and X′′ = X we find that the spectral sequence for X is a module (in the appropriate sense) over that for the sphere S^{0}. □

The second result is

2.3.4. Theorem. Let E_{∗}(h) = 0 in 2.3.2. Then for 2 ≤ r ≤ ∞ there are maps δ_{r}: E_{r}^{s,∗}(Y) → E_{r}^{s+1,∗}(W) such that

(a) δ_{2} is the connecting homomorphism associated with the short exact sequence

0 → E_{∗}(W) → E_{∗}(X) → E_{∗}(Y) → 0,

(b) δ_{r}d_{r} = d_{r}δ_{r} and δ_{r+1} induced by δ_{r},

(c) δ_{∞} is a filtered form of the map π_{∗}(h). The connecting homomorphism in Ext can be described as the Yoneda product (Hilton and Stammbach [1, p. 155] with the element of Ext_{E_{∗}(E)}^{1}(E_{∗}(Y), E_{∗}(W))

corresponding to the short exact sequence

0 → E_{∗}(W) → E_{∗}(X) → E_{∗}(Y) → 0.

Similarly, given a sequence of maps

X_{0} ⟶^{f_{0}} ΣX_{1} ⟶^{f_{1}} Σ^{2}X_{2} → ⋯ → Σ^{n}X_{n}

with E_{∗}(f_{i}) = 0 one gets maps

δ_{r}: E_{r}^{s,∗}(X_{0}) → E_{r}^{s+n∗}(X_{n})

commuting with differentials where δ_{2} can be identified as the Yoneda product with the appropriate element in

Ext_{E(E_{∗})}^{n}(E_{∗}(X_{0}), E_{∗}(X_{n})).  □

If one generalizes the spectral sequence to source spectra other than the sphere one is led to a pairing induced by composition of maps. This has been studied by Moss [1], where it is assumed that one has Adams resolutions satisfying much stronger conditions than 2.2.1. In the spectral sequence for the sphere it can be shown that the composition and smash product pairings coincide, but we will not need this fact.

To prove 2.3.3 we will use the canonical resolutions (2.2.9) for X′, X′′ and X. Recall that these can be obtained by smashing the respective spectra with the canonical resolution for S^{0}. Let K_{s,s+r} be the cofiber in

(2.3.5)  \overline{E}^{(s+r)} → \overline{E}^{(s)} → K_{s,s+r},

where E is the fiber of S^{0} → E.

These spectra have the following properties.

2.3.6. Lemma.

(a) There are canonical fibrations

K_{s+i,s+i+j} → K_{s,s+i+j} → K_{s,s+i}.

(b) E_{1}^{s,∗}(X) = π_{∗}(X ∧ K_{s,s+1}).

Let Z_{r}^{s,∗}(X), B_{r}^{s,∗}(X) ⊂ E_{1}^{s,∗}(X) be the images of π_{∗}(X ∧ K_{s,s+r}) and π_{∗}(X ∧ Σ^{−1}K_{s−r+1,s}), respectively. Then E_{r}^{s,∗}(X) = Z_{r}^{s,∗}(X)/B_{r}^{s,∗}(X) and d_{r} is induced by the map

X ∧ K_{s,s+r} → X ∧ ΣK_{s+r,s+2r}.

(c) α induces map X′_{s}∧X′′_{t} → X_{s+1} (where these are the spectra in the canonical resolutions) compatible with the maps g′_{s}, g′′_{t}, and g_{s+t} of 2.2.1.

(d) The map

K_{s,s+1} ∧ K_{t,t+1} → K_{s+t,s+t+1},

given by the equivalence

K_{n,n+1} = E ∧ \overline{E}^{(n)}

and the multiplication on E, lifts to maps

K_{s,s+r} ∧ K_{t,t+r} → K_{s+t,s+t+r}

for r > 1 such that the following diagram commutes

\xymatrix{K_{s,s+r+1} ∧ K_{t,t+r+1} \ar[r] \ar[d] & K_{s+t,s+t+r+1} \ar[d] \\ K_{s,s+r} ∧ K_{t,t+r} \ar[r] & K_{s+t,s+t+r}}

where the vertical maps come from (a).

(e) The following diagram commutes

\xymatrix{K_{s,s+r} ∧ K_{t,t+r} \ar[rr] \ar[d] &  & (ΣK_{s+r,s+2r} ∧ K_{t,t+r}) ∨ (K_{s,s+r} ∧ ΣK_{t+r,t+2r}) \ar[dl] \\ K_{s+t,s+t+r} \ar[r] & ΣK_{s+t+r,s+t+2r}}

where the vertical maps are those of (d) and the horizontal maps come from (a), the maps to and from the wedge being the sums of the maps to and from the summands.

Proof. Part (a) is elementary. For (b) we refer the reader to Cartan and Eilenberg [1], Section XV.7, where a spectral sequence is derived from a set of abelian groups H(p, q) satisfying certain axioms. Their H(p, q) in this case is our π_{∗}(K_{p,q}), and (a) guarantees that these groups have the appropriate properties. For (c) we use the fact that X′_{s} = X′ ∧ \overline{E}^{(s)}, X′′_{t} = X′′ ∧ \overline{E}^{(t)}, and X_{s+t} = X ∧ \overline{E}^{(s+t)}.

For (d) we can assume the maps \overline{E}^{(s+1)} → \overline{E}^{(s)} are all inclusions with K_{s,s+r} = \overline{E}^{(s)}/\overline{E}^{(s+r)}. Hence we have

K_{s,s+r} ∧ K_{t,t+r} = E^{(s)} ∧ \overline{E}^{(t)}/(\overline{E}^{(s+r)} ∧ \overline{E}^{(t)} ∪ \overline{E}^{(s)} ∧ \overline{E}^{(t+r)})

and this maps naturally to

\overline{E}^{(s+t)}/\overline{E}^{(s+t+r)} = K_{s+t,s+t+r}.

For (e) if \overline{E}^{(s+2r)} → \overline{E}^{(s+r)} → \overline{E}^{(s)} are inclusions then so is K_{s+r,s+2r} → K_{s,s+2r} so we have K_{s,s+r} = K_{s,s+2r}/K_{s+r,s+2r} and K_{t,t+r} = K_{t,t+2r}/K_{t+r,t+2r}. With this in mind we get a commutative diagram

\xymatrix{K_{s,s+r} ∧ K_{t+r,t+2r} ∪ K_{s+r,s+2r} ∧ K_{t,t+r} \ar[r] \ar[d] & K_{s+t+r,s+t+2r} \ar[d] \\ K_{s,s+2r} ∧ K_{t,t+2r} \ar[d] \ar[r] & K_{s+t,s+t+2r} \ar[d] \\ K_{s,s+r} ∧ K_{t,t+r} \ar[r] \ar[d] & K_{s+t,s+t+r} \ar[d] \\ Σ(K_{s,s+r} ∧ K_{t+r,t+2r} ∪ K_{s+r,s+2r} ∧ K_{t,t+r}) \ar[r] & ΣK_{s+t+r,s+t+2r}}

where the horizontal maps come from (d) and the upper vertical maps are inclusions. The lower left-hand map factors through the wedge giving the desired diagram. □

We are now ready to prove 2.3.3. In light of 2.3.6(b), the pairing is induced by the maps of 2.3.6(d). Part 2.3.3(a) then follows from 2.3.6(e) as the differential on E_{r}^{∗∗}(X′)⊗E_{r}^{∗∗}(X′′) is induced by the top map of 2.3.6(e). Part 2.3.3(b) follows from the commutative diagram in 2.3.6(d). Part 2.3.3(c) follows from the compatibility of the maps in 2.3.6(c) and (d).

Assuming 2.3.3(e), (d) is proved as follows. The pairing on Ext is functorial, so if E_{∗}(X) has a product which is associative or commutative, so will E_{2}^{∗∗}(X). Now suppose inductively that the product on E_{r}^{∗∗}(X) has the desired property. Since the product on E_{r+1} is induced by that on E_{r} the inductive step follows.

It remains then to prove 2.3.3(e). We have E_{∗}(X′ ∧ K_{s,s+1}) = D^{s}(E_{∗}(X′)) (A1.2.11) and similarly for X′′, so our pairing is induced by a map

E_{∗}(X′ ∧ K_{s,s+1}) ⊗_{π_{∗}(E)} E_{∗}(X′′ ∧ K_{t,t+1}) → E_{∗}(X ∧ K_{s+t,s+t+1}),

i.e., by a pairing of resolutions. Hence the pairing on E_{2} coincides with the specified algebraic pairing by the uniqueness of the latter (A1.2.14).

We prove 2.3.4 by reducing it to the following special case.

2.3.7. Lemma. Theorem 2.3.4 holds when X is such that Ext^{s}(E_{∗}(X)) = 0 for s > 0 and π_{∗}(X) = Ext^{0}(E_{∗}(X)). □

Proof of 2.3.4. Let W′ be the fiber of the composite

W ⟶^{f} X → X ∧ E.

Since Σfh is trivial, h lifts to a map h′: Y → ΣW′. Now consider the cofiber sequence

W → X ∧ E → ΣW′ → ΣW.

Lemma 2.3.7 applies here and gives maps

δ_{r}: E_{r}^{s,∗}(ΣW′) → E_{r}^{s+1,∗}(ΣW).

Composing this with the maps induced by h′ gives the desired result. □

Proof of 2.3.7. Disregarding the notation used in the above proof, let W′ = Σ^{−1}Y, X′ = Σ^{−1}Y ∧ E, and Y′ = Y ∧ \overline{E}. Then we have a commutative diagram in which both rows and columns are cofiber sequences

\xymatrix{X \ar[d] & W \ar[l] \ar[d] & W′ \ar[d] \ar[l] \\ X ∨ (Y ∧ E) \ar[d] & X \ar[l] \ar[d] & X′ \ar[l] \ar[d] \\ Y ∧ E & Y \ar[l] & Y′ \ar[l]}

Each row is the beginning of an Adams resolution (possibly noncanonical for W and X) which we continue using the canonical resolutions (2.2.9) for W′, X′, and Y′. Thus we get a commutative diagram

(2.3.8)  \xymatrix{W \ar[d] & W′ \ar[l] \ar[d] & W′ ∧ \overline{E} \ar[l] \ar[d] & W′ ∧ \overline{E}^{(2)} \ar[d] \ar[l] & ⋯ \ar[l] \\ X \ar[d] & X′ \ar[l] \ar[d] & X′ ∧ \overline{E} \ar[d] \ar[l] & X′ ∧ \overline{E}^{(2)} \ar[d] \ar[l] & ⋯ \ar[l] \\ Y & Y′ \ar[l] & Y′ ∧ \overline{E} \ar[l] & Y′ ∧ \overline{E}^{(2)} \ar[l] & ⋯ \ar[l]}

in which each column is a cofiber sequence. The map Y ⟶^{≃} ΣW′ induces maps δ_{r}: E_{r}^{s,∗}(Y) → E_{r}^{s+1,∗}(W) which clearly satisfy 2.3.4(a) and (b), so we need only to verify that δ_{2} is the connecting homomorphism. The resolutions displayed in 2.3.8 make this verification easy because they yield a short exact sequence of E_{1}-terms which is additively (though not differentially) split. For s = 0 we have

\begin{aligned}E_{1}^{0,∗}(W) & = π_{∗}(X), E_{1}^{0,∗}(X) = π_{∗}(X ∨ (Y ∧ E)), \\ E_{1}^{0,∗}(Y) & = π_{∗}(Y ∧ E), E_{1}^{1,∗}(W) = π_{∗}(Y ∧ E), \\ E_{1}^{1,∗}(X) & = π_{∗}(Y ∧ E ∧ E) and E_{1}^{1,∗}(Y) = π_{∗}(ΣY ∧ \overline{E} ∧ E), \end{aligned}

so the relevant diagram for the connecting homomorphism is

\xymatrix{X \ar[r] \ar[d] & X ∨ (Y ∧ E) \ar[r] \ar[d]_{d} & Y ∧ E \ar[l]_{b} \ar[d] \\ Y ∧ E \ar[r] & Y ∧ E ∧ E \ar[l]_{a} \ar[r] & ΣY ∧ \overline{E} ∧ \overline{E}}

where a and b are splitting maps. The connecting homomorphism is induced by adb, which is the identity on Y ∧ E, which also induces δ_{2}.

For s > 0 we have

E_{1}^{s,∗}(W) = π_{∗}(Σ^{s−1}Y ∧ E ∧ \overline{E}^{(s−1)}),
E_{1}^{s,∗}(X) = π_{∗}(Σ^{s−1}Y ∧ E^{(2)} ∧ \overline{E}^{(s−1)}),

and

E_{1}^{s,∗}(Y) = π_{∗}(Σ^{s}Y ∧ E ∧ \overline{E}^{(s)}),

so the relevant diagram is

\xymatrix{E \ar[r] \ar[d] & E ∧ E \ar[r] \ar[d] & E ∧ \overline{E} \ar[l] \ar[d] \\ ΣE ∧ \overline{E} \ar[r] & ΣE ∧ E ∧ \overline{E} \ar[l] \ar[r] & Σ^{2}E ∧ \overline{E}^{2}}

and again the connecting homomorphism is induced by the identity on Σ^{s}Y ∧ E ∧ \overline{E}^{s}. □