THE TRIPLE LOOP SPACE APPROACH TO THE TELESCOPE CONJECTURE

MARK MAHOWALD, DOUGLAS RAVENEL AND PAUL SHICK

September 11, 2000

Contents

1. The telescope conjecture and Bousfield localization 3 1.1. Telescopes 3 1.2. Bousfield localization and Bousfield classes 5 1.3. The telescope conjecture 7 1.4. Some other open questions 9 2. Some variants of the Adams spectral sequence 9 2.1. The classical Adams spectral sequence 10 2.2. The Adams-Novikov spectral sequence 12 2.3. The localized Adams spectral sequence 14 2.4. The Thomified Eilenberg-Moore spectral sequence 19 2.5. Hopf algebras and localized Ext groups 22 3. The spectra y(n) and Y (n) 25 3.1. The EHP sequence and some Thom spectra 25 3.2. The homotopy of L_{n}y(n) and Y (n) 29 3.3. The triple loop space 33 4. Properties of Ω^{3}S^{1+2p^{n}} 35 4.1. The Snaith splitting 35 4.2. Ordinary homology 36 4.3. Morava K-theory 40 4.4. The computation of Y (n)_{∗}(Ω^{3}S^{1+2p^{n}}) via the Eilenberg-

Moore spectral sequence 45

5. Toward a proof of the differentials conjecture 49 5.1. The E_{2}-term of the localized Thomified Eilenberg-Moore

spectral sequence 49

5.2. Short differentials 56 5.3. Excluding spurious differentials 61 References 65

The purpose of this paper is to describe an unsuccessful attempt to prove that the telescope conjecture (see 1.13 below for the precise statement) is false for n ≥ 2 and each prime p. At the time the it was originally formulated over 20 years ago (see [Rav84]), the telescope conjecture appeared to be the simplest and most plausible statement about the question at hand, namely the relation between two different localization functors. We hope the present paper will demonstrate that this is no longer the case. We will set up a spectral sequence converging to the homotopy of one of the two localizations (the geometrically defined telescope) of a certain spectrum, and it will be apparent that only a bizarre pattern of differentials would lead to the known homotopy of the localization defined in terms of BP -theory, the answer predicted by the telescope conjecture. While we cannot exclude such a pattern, it is certainly not favored by Occam’s razor.

No use will be made here made of the parametrized Adams spectral sequence of [Rav92b]; we will say more about that approach in a future paper. Instead we will rely on some constructions related to the EHP sequence which are described in §3, where we define the spectra y(n) and Y (n), and a variant of the Eilenberg-Moore spectral sequence (which we call the Thomified Eilenberg-Moore spectral sequence) described in §2.4.

§1 is an expository introduction to the telescope conjecture. We define telescopes and recall the nilpotence (1.1), periodicity (1.4) and thick subcategory (1.12) theorems of Devinatz, Hopkins and Smith ([DHS88] and [HS98]). We also recall the definitions of Bousfield localization and related concepts and the Bousfield localization theorem (1.8). We then state four equivalent formulations of the telescope conjecture in 1.13.

In §2 we introduce the various spectral sequences that we will use. These include the classical Adams (§2.1) and Adams-Novikov (§2.2) spectral sequences. We also need the localized Adams spectral sequence of Miller [Mil81] (§2.3), for which we prove a convergence theorem 2.13. This is the spectral sequence we will use to compute the homotopy of our telescope Y (n) and see that it may well differ from the answer predicted by the telescope conjecture. In §2.4 we introduce the Thomified Eilenberg-Moore spectral sequence and its localized form. In certain cases (2.26 and 2.27) we identify its E_{2}-term as Ext over a Massey-Peterson algebra. All of these spectral sequences require the use of Ext groups over various Hopf algebras, and we review the relevant homological algebra in §2.5. This includes two localizations ((2.34) and (2.35)) of the Cartan-Eilenberg spectral sequence which are new as far as we know.

In §3 we use the EHP sequence to construct the spectrum y(n) and its telescope Y (n). We describe the computation of π_{∗}(L_{n}y(n)) using the Adams-Novikov spectral sequence, and then state our main computational conjecture, 3.16, which says that the localized Adams spectral sequence gives a different answer for π_{∗}(Y (n)) when n > 1. This would disprove the telescope conjecture, which predicts that L_{n}y(n) = Y (n). The conjectured difference between π_{∗}(L_{n}y(n)) and π_{∗}(Y (n)) can be described very simply: π_{∗}(L_{n}y(n)) is finitely generated as a module over the ring K(n)_{∗}[v_{n+1}, v_{n+2}, …, v_{2n}], whereas, if our main conjecture is correct, then π_{∗}(Y (n)) will have no finite presentation over this ring.

Our construction of y(n) gives us a map

Ω^{3}S^{1+2p^{n}} ⟶^{f} y(n),

with which we originally hoped to prove Conjecture 3.16 and is the reason for the title of this paper. In §4 we recall some properties Ω^{3}S^{1+2p^{n}}, including the Snaith splitting (4.2) and its ordinary homology as a module over the Steenrod algebra (Lemma 4.7). In §4.3 we recall Tamaki’s unpublished computation of its Morava K-theory using his formulation [Tam94] of the Eilenberg-Moore spectral sequence, and in §4.4 we show that similar methods can be used to compute its Y (n)_{∗}-theory. These are not needed for our main results and are included due to their independent interest.

In §5 we describe our program to prove Conjecture 3.16 and thereby disprove the telescope conjecture for n > 1. Our method is to construct a map (derived from the map f above) to the localized Adams spectral sequence for Y (n)_{∗} from a localized Thomified Eilenberg-Moore spectral sequence converging to Y (n)_{∗}(Ω^{3}S^{1+2p^{n}}). This map turns out to be onto in each E_{r}, so differentials in the latter spectral sequence are determined by those in the former, which are described in Conjecture 5.15. The source spectral sequence has far more structure than the target, and we had hoped to use this to prove 5.15. There are three such structures, each of which figures in the program, namely:

(i) Ω^{3}S^{1+2p^{n}} is an H-space, so the spectral sequence is one of Hopf algebras.

(ii) It has a Snaith splitting which must be respected by differentials. (iii) The pth Hopf map induces an endomorphism of our spectral se-

quence, which is identified in Lemma 5.16.

Previously we had thought that this structure could be used to construct certain permanent cycles ỹ_{i,j} mapping to b_{n+i,j} in the localized Adams spectral sequence that would force the latter to collapse from a certain stage. Unfortunately, this is not the case. For more details, see the comments after Conjecture 5.12.

It is pleasure to acknowledge helpful conversations with Fred Cohen, Bill Dwyer, Emmanuel Dror Farjoun, Mike Hopkins, Nick Kuhn, Haynes Miller, Hal Sadofsky, Brooke Shipley, and Dai Tamaki. We also thank the referee for many helpful suggestions.

1. The telescope conjecture and Bousfield localization

1.1. Telescopes. The telescope conjecture is a statement about the stable homotopy groups of finite complexes. There is not a single nontrivial example for which such groups are completely known. There are many partial results, especially about the stable homotopy groups of spheres. Unstably the situation is only slightly better. We have complete knowledge of π_{∗}(X) for a finite complex X only in the cases where X is known to be an Eilenberg-Mac Lane space, such as when X is a surface of positive genus.

Experience has shown that one can get interesting information about π_{∗}(X) in the stable case in the following way. Suppose one has a stable map of the form

Σ^{d}X ⟶^{f} X

for which all iterates are essential; this can only happen if d ≥ 0. Such a map is said to be periodic. We say that f is nilpotent if some iterate of it is null. In any case we can define the telescope f^{−1}X to be the direct limit of the system

X ⟶^{f} Σ^{−d}X ⟶^{f} Σ^{−2d}X ⟶^{f} · · · .

This will be contractible if f is nilpotent. In the (rare) cases when f is periodic, the computation of π_{∗}(f^{−1}X) is far more tractable than that of π_{∗}(X).

The map f induces an endomorphism of π_{∗}(X), which we will denote abusively by f, making π_{∗}(X) a module over the ring Z[f]. Since homotopy commutes with direct limits, we have

π_{∗}(f^{−1}X) = π_{∗}(X) ⊗_{Z[f]} Z[f, f^{−1}].

The telescope conjecture is a statement about this graded group.

Before stating it we will describe some motivating examples. We assume that all spaces and spectra in sight are localized at a prime p.

• For any spectrum X let f be the degree p map. It induces multiplication by p in homotopy and homology and induces an isomorphism in rational homology. If H_{∗}(X; Q) is nontrivial, i.e., if the integer homology of X is not all torsion, then all iterates of the degree p map are essential.

In this case the telescope p^{−1}X is the rationalization XQ of X with

π_{∗}(XQ) = π_{∗}(X) ⊗ Q = H_{∗}(X; Q),

the rational homotopy of X. It is a rational vector space.

• Let V (0) be the mod p Moore spectrum. For each prime p Adams [Ada66] constructed a map

Σ^{d}V (0) ⟶^{α} V (0) where d = \{ \begin{matrix}8 & if p = 2 \\ 2p − 2 & if p is odd. \end{matrix}

This map induces an isomorphism in classical K-theory and all iterates of it are nontrivial. π_{∗}(α^{−1}V (0)) has been computed explicitly by Mahowald [Mah81] for p = 2 and Miller [Mil81] for odd primes. It is finitely presented as a module over Z[α, α^{−1}]. The image of π_{∗}(V (0)) in π_{∗}(α^{−1}V (0)) is known, and this gives us a lot of information about the former.

By analogy with the previous example, one might expect π_{∗}(α^{−1}V (0)) to be K_{∗}(V (0)), but the situation here is not so simple. The answer is however predictable by K-theoretic or BP-theoretic methods; we will say more about this later.

• For odd p let V (1) denote the cofiber of the Adams map α. It is a CW-complex with one cell each in dimensions 0, 1, 2p − 1 and 2p. Smith [Smi71] and Toda [Tod71] have shown that for p ≥ 5 there is a periodic map

Σ^{2p^{2}−2}V (1) ⟶^{β} V (1).

In this case the homotopy of the telescope is not known. The results of Devinatz-Hopkins-Smith ([DHS88] and [HS98]) allow

us to study telescopes in a very systematic way. They indicate that BP-theory and Morava K-theory are very useful here. First we have the nilpotence theorem characterizing nilpotent maps.

Theorem 1.1 (Nilpotence theorem). For a finite p-local spectrum X, a map

Σ^{d}X ⟶^{f} X

is nilpotent if and only if the induced map on BP_{∗}(X) is nilpotent. Equivalently, it is nilpotent if and only if the induced map on K(n)_{∗}(X) is nilpotent for each n.

For the study of periodic maps two definitions are useful.

Definition 1.2. A p-local finite complex X has type n if n is the smallest integer for which K(n)_{∗}(X) is nontrivial.

Definition 1.3. A map

Σ^{d}X ⟶^{f} X

is a v_{n}-map if K(n)_{∗}(f) is an isomorphism and K(m)_{∗}(f) = 0 for m ≠ n. (The spectrum X here need not be finite.)

A finite complex of type n does not admit a v_{m}-map for m > n; this follows from the algebraic properties of the target category of the BP-homology functor. For m < n, the trivial map is a v_{m}-map. The cofiber of a v_{n}-map on a type n complex is necessarily a complex of type n + 1. In the three examples above we have a such a map for n = 0, 1 and 2 respectively.

Now we can state the periodicity theorem of [HS98].

Theorem 1.4 (Periodicity theorem). Every type n finite complex admits a v_{n}-map. Given two such maps f and g there are positive integers i and j such that f^{i} = g^{j}.

Corollary 1.5. For a type n p-local finite complex X, any v_{n}-map f : Σ^{d}X → X yields the same telescope f^{−1}X, which we will denote by v_{n}^{−1}X or X̂.

1.2. Bousfield localization and Bousfield classes.

Definition 1.6. Given a homology theory h_{∗}, a spectrum X is h_{∗}-local if for each spectrum W with h_{∗}(W) = 0, [W, X] = 0. An h_{∗}-localization X → L_{h}X is an h_{∗}-equivalence from X to an h_{∗}-local spectrum. We denote the fiber of this map by C_{h}X. If h_{∗} is represented by a spectrum E we will write L_{E} and C_{E} for L_{h} and C_{h}. The case E = v_{n}^{−1}BP is of special interest, and we denote the corresponding functors by L_{n} and C_{n}.

The following properties of localization are formal consequences of these definitions.

Proposition 1.7. If L_{h}X exists it is unique and the functor L_{h} is idempotent. The map X → L_{h}X is terminal among all h_{∗}-equivalences from X and initial among all maps from X to h_{∗}-local spectra. C_{h}X is h_{∗}-acyclic and the map C_{h}X → X is terminal among all maps from h_{∗}-acyclics to X. The homotopy inverse limit of h_{∗}-local spectra is h_{∗}-local, although the functor L_{h} (if it exists) need not commute with homotopy inverse or direct limits. The homotopy direct limit of local spectra need not be local.

The definitive theorem in this subject is due to Bousfield [Bou79].

Theorem 1.8 (Bousfield localization theorem). The localization L_{h}X exists for all spectra X and all homology theories h_{∗}.

Roughly speaking, one constructs C_{h}X by taking the direct limit of all h_{∗}-acyclic spectra mapping to X. (This is not precisely correct because of set theoretic problems; there are too many such maps to form a direct limit. Bousfield found a way around this difficulty.) A variant on this procedure is to consider the homotopy direct limit of all finite h_{∗}-acyclic spectra mapping to X, which we denote by C_{h}^{f}X. (Here f stands for finite, and there are no set theoretic problems.) We denote the cofiber of C_{h}^{f}X → X by L_{h}^{f}X.

Definition 1.9. A localization functor L_{h} is finite if L_{h} = L_{h}^{f}, i.e., if C_{h}X is always a homotopy direct limit of finite h_{∗}-acyclic spectra mapping to X.

Proposition 1.10. If the functor L_{h} is finite then

(i) it commutes with homotopy direct limits, (ii) the homotopy direct limit of h_{∗}-local spectra is local, (iii) L_{h}X = X ∧ L_{h}S^{0} for all X, and (iv) L_{h} is the same as Bousfield localization with respect to the homol-

ogy theory represented by L_{h}S^{0}.

It can be shown [Rav84, Prop. 1.27] that the four properties listed in 1.10 are equivalent. We say that a localization functor is smashing if it has them. Thus 1.10 says that every finite localization functor is smashing. Bousfield conjectured [Bou79, 3.4] the converse, that every smashing localization functor is finite. The functor L_{n} is known to be smashing [Rav92a, Theorem 7.5.6], but if the telescope conjecture fails, it is not finite for n ≥ 2.

Definition 1.11. Two spectra E and F are Bousfield equivalent if they have the same acyclics (i.e. if E_{∗}(X) = 0 iff F_{∗}(X) = 0,) or equivalently if L_{E} = L_{F}. The corresponding equivalence class is denoted by ⟨E⟩, the Bousfield class of E. We say that ⟨E⟩ ≥ ⟨F ⟩ if E_{∗}(X) = 0 implies F_{∗}(X) = 0.

Dror Farjoun [Far96] uses the notation X ≪ Y (Y can be built from X by cofibrations) in an unstable context to mean ⟨X⟩ > ⟨Y ⟩.

The following consequence of 1.1 is very useful, e.g. it was used to prove 1.4. A subcategory of the stable homotopy category of finite complexes is thick if it is closed under cofibrations and retracts. One example is the subcategory of h_{∗}-local finite spectra for a given h. The following result of [DHS88] classifies all thick subcategories.

Theorem 1.12 (Thick subcategory theorem). Any nontrivial thick subcategory of the stable homotopy category of p-local finite complexes is the category C_{n} of p-local finite K(n − 1)_{∗}-acyclic spectra for some n ≥ 0.

Note that C_{0} is the entire category of p-local finite spectra,

C_{0} ⊃ C_{1} ⊃ C_{2} ⊃ …,

and the intersection of all these is the trivial subcategory consisting of a point.

This dry sounding theorem is a useful tool. Suppose one wants to prove that all p-local finite spectra of type ≥ n satisfy a certain property, say that they are all demented. (This example is due to John Harper.) If one can show that the subcategory of demented spectra is thick, then all that remains is to show that a single one of type n is demented. If one is demented they all are demented. Conversely, if we can find a single type n spectrum that is not demented, then none of them are.

1.3. The telescope conjecture. Now we will discuss several equivalent formulations of the telescope conjecture.

Telescope conjecture 1.13. Choose a prime p and an integer n ≥ 0. Let X be a p-local finite complex of type n (1.2) and let X̂ be the associated telescope (1.5). Then

(i) X̂ = L_{n}X. (ii) ⟨X̂⟩ = ⟨K(n)⟩. (iii) The Adams-Novikov spectral sequence for X̂ converges to π_{∗}(X̂). (iv) The functors L_{n} and L_{n}^{f} are the same if L_{n−1} = L_{n−1}^{f}.

We will sketch the proof that the four statements above are equivalent.

The set of K(n − 1)_{∗}-acyclic finite p-local spectra satisfying (i) is thick. The same is true for (ii) and for the statement that

⟨L_{n}X⟩ ≤ ⟨K(n)⟩. (1.14)

Thus if we can find a type n X with this property it will follow that (i) and (ii) are equivalent. One can show that (1.14) holds if the Adams-Novikov E_{2}-term for X has a horizontal vanishing line; this means that L_{n}X can be built out of K(n) with a finite number of cofibrations. Such an X can be constructed using the methods described in [Rav92a, §8.3].

For the third statement, the Adams-Novikov spectral sequence for L_{n}X (which is BP_{∗}-equivalent to X̂) was shown in [Rav87] to converge to its homotopy, so it also converges to that of X̂ iff (i) holds.

For the fourth statement, since the functors L_{n} and L_{n}^{f} are both smashing, they commute with homotopy direct limits. This means that if they agree on finite complexes, they agree on all spectra. For K(n − 1)_{∗}-acyclic X it is known that L_{n}^{f}X = X̂ (see [Rav93b], Miller [Mil92] or Mahowald-Sadofsky [MS95]) so (i) says the two functors agree on such X. For finite p-local X of smaller type, the methods of [Rav93b, §2] show that C_{n−1}^{f} (the fiber of X → L_{n−1}^{f}X) is a homotopy direct limit of type n finite complexes, so L_{n}X = L_{n}^{f}X.

Any attempt to prove 1.13 is likely to rely on 1.12. It is easy to show that the set of K(n − 1)_{∗}-acyclic finite spectra satisfying 1.13(i) is thick. Thus one can prove or disprove the telescope conjecture if we can compare π_{∗}(L_{n}X) with π_{∗}(X̂) for a single type n spectrum X. The telescope conjecture for n = 1 follows from the computations of Mahowald [Mah81] and Miller [Mil81] of π_{∗}({ \atop V (0)}) which showed that agrees with the previously known value of π_{∗}(L_{1}V (0)). Alternately we can disprove the telescope conjecture by finding a spectrum Y (which need not be finite) for which L_{n}^{f}Y ≠ L_{n}Y.

The groups π_{∗}(L_{n}X) (or π_{∗}(L_{n}Y)) and π_{∗}(X̂) (or π_{∗}(L_{n}^{f}Y)) can be computed with variants of the Adams spectral sequence. These methods will be discussed in the next section.

The spectrum we will use, y(n), is a certain Thom spectrum which will be constructed in §3. We will use the Adams-Novikov spectral sequence to show (Corollary 3.12) that π_{∗}(L_{n}y(n)) is finitely generated over a certain ring R(n)_{∗} defined below in (3.13); this is relatively easy. A far more difficult calculation (Conjecture 3.16) using the localized Adams spectral sequence (described in §2.3) comes quite close to showing that π_{∗}(L_{n}^{f}y(n)) is not finitely generated over R(n)_{∗} for n > 1, which would disprove the telescope conjecture.

1.4. Some other open questions. The spectrum y(n) of §3 has a telescope Y (n) associated with it. Conjecture 3.9 below says that 1.13(ii) holds with K(n) replaced by Y (n). Computing π_{∗}(Y (n)) is the main object of this paper. Each Y (n) is a module over a spectrum T_{∞} (4.3) and we suspect (4.4) that T_{∞} has the same Bousfield class as the sphere spectrum.

The functors L_{n} could be called chromatic localizations. There are natural transformations from L_{n+1} to L_{n}, so for each spectrum X we have an inverse system

L_{0}X ⟵ L_{1}X ⟵ L_{2}X ⟵ ⋯,

and we can ask if the natural map from X to the homotopy inverse limit is an equivalence. This is the chromatic convergence question. The chromatic convergence theorem of Hopkins and the author [Rav92a, 7.5.7] says that this is the case for p-local finite spectrum X.

The telescopic convergence question concerns the inverse limit of the L_{n}^{f}X, its telescopic localizations. We know that there are maps

X ⟶ L_{n}^{f}X ⟶ L_{n}X

and that holim L_{n}X ≃ X, so X is a retract of holim L_{n}^{f}X. It suffices to answer this question for the case X = S^{0}, since L_{n}^{f}X = X ∧ L_{n}^{f}S^{0} (L_{n}^{f} is smashing) and smashing with a finite complex preserves inverse limits.

2. Some variants of the Adams spectral sequence

The Adams spectral sequence for π_{∗}(X) is derived from the following Adams diagram.

X_{0}  X_{1}  X_{2}  ⋯ K_{0}  K_{1}  K_{2}_{_{g_{0}} u}  _{_{g_{1}} u}_{u}  _{_{g_{2}} u}_{u}  _{u} (2.1)

Here X_{s+1} is the fiber of g_{s}. We get an exact couple of homotopy groups and a spectral sequence with

E_{1}^{s,t} = π_{t−s}(K_{s}) and d_{r} : E_{r}^{s,t} → E_{r}^{s+r,t+r−1}.

This spectral sequence converges to π_{∗}(X) if the homotopy inverse limit lim_{←} X_{s} is contractible and certain lim^{1} groups vanish. When X is connective, the Adams spectral sequenceis generally displayed like a first quadrant spectral sequence. For more background, see [Rav86].

Now suppose we have a generalized homology theory represented by a ring spectrum E. Then the canonical E-based Adams resolution for X is the diagram (2.1) with K_{s} = E ∧ X_{s}. More generally an E-based Adams resolution for X is such a diagram where K_{s} is such that the map g_{s} ∧ E is the inclusion of a retract. Under certain hypotheses on E the resulting E_{2}-term is independent of the choice of resolution and can be identified as an Ext group. The classical Adams spectral sequence is the case where E = H/p, the mod p Eilenberg-Mac Lane spectrum, and the Adams-Novikov spectral sequence is the case where E = BP, the Brown-Peterson spectrum. We will have occasion to use a noncanonical Adams resolution below for a case where E = H/p. Then the condition on the diagram is that H_{∗}(g_{s}) be monomorphic for each s.

2.1. The classical Adams spectral sequence. Here we have

E_{2}^{s,t} = Ext_{A_{∗}}(Z/(p), H_{∗}(X)),

where A_{∗} is the dual Steenrod algebra, H_{∗}(X) is the mod p homology of X, and Ext is taken in the category of A_{∗}-comodules. This group is the same as Ext_{A}(H^{∗}(X), Z/(p)), where H^{∗}(X) is regarded as a module over the Steenrod algebra A. This group is not easy to compute in most cases. There is not a single nontrivial example where X is finite and this group is completely known, although there are good algorithms for computing it in low dimensions.

We recall the structure of A_{∗}. When working over a field k we will use the notation P (x) and E(x) to denote polynomial and exterior algebras over k on x. As an algebra we have

A_{∗} = \{ \begin{matrix}P (ξ_{1}, ξ_{2}, …) & with |ξ_{i}| = 2^{i} − 1 \\ & for p = 2 \\ P (ξ_{1}, ξ_{2}, …) ⊗ E(τ_{0}, τ_{1}, …) & with |ξ_{i}| = 2p^{i} − 2 \\ & and |τ_{i}| = 2p^{i} − 1 \\ & for p > 2. \end{matrix}

For odd primes we will denote the polynomial and exterior factors by P_{∗} and Q_{∗} respectively. For p = 2, P_{∗} and Q_{∗} will denote P (ξ_{i}^{2}) and E(ξ_{i}) respectively. The coproduct is given by

Δ(ξ_{i}) = ∑_{0≤j≤i} ξ_{i−j}^{p^{j}} ⊗ ξ_{j}  where ξ_{0} = 1.
and Δ(τ_{i}) = τ_{i} ⊗ 1 + ∑_{0≤j≤i} ξ_{i−j}^{p^{j}} ⊗ τ_{j}.

In §2.5 we will review some facts about Ext groups over Hopf algebras such as A_{∗}, which we will refer to here when needed.

In §3 we will construct a spectrum y(n) with

H_{∗}(y(n)) = \{ \begin{matrix}P (ξ_{1}, …, ξ_{n}) & for p = 2 \\ P (ξ_{1}, …, ξ_{n}) ⊗ E(τ_{0}, …, τ_{n−1}) & for p > 2. \end{matrix}

Let

B(n)_{∗} = \{ \begin{matrix}A_{∗}/(ξ_{1}, …, ξ_{n}) & for p = 2 \\ A_{∗}/(ξ_{1}, …, ξ_{n}, τ_{0}, …, τ_{n−1}) & for p > 2 \end{matrix} (2.2)

Then we have H_{∗}(y(n)) = A_{∗}□_{B(n)_{∗}}Z/(p), and we can use the changeof-rings isomorphism (2.30) to prove

Proposition 2.3. With notation as above

Ext_{A_{∗}}(Z/(p), H_{∗}(y(n))) = Ext_{B(n)_{∗}}(Z/(p), Z/(p)).

For future reference we record some information about this Ext group. For a fixed value of n let

(2.4)  P′_{∗} = \{ \begin{matrix}P (ξ_{n+1}, ξ_{n+2}, …) & for p > 2 \\ P (ξ_{n+1}^{2}, ξ_{n+2}^{2}, …) & for p = 2 \end{matrix}
(2.5)  Q′_{∗} = \{ \begin{matrix}E(τ_{n}, τ_{n+1}, …) & for p > 2 \\ E(ξ_{n+1}, ξ_{n+2}, …) & for p = 2 \end{matrix}

Then we have a Hopf algebra extension (2.31)

P′_{∗} ⟶ B(n)_{∗} ⟶ Q′_{∗} (2.6)

and a Cartan-Eilenberg spectral sequence (2.32) converging to the group of 2.3 with

E_{2} = Ext_{P′_{∗}}(Z/(p), Ext_{Q′_{∗}}(Z/(p), Z/(p)))
= Ext_{P′_{∗}}(Z/(p), V′).

where

V′ = P (v_{n}, v_{n+1}, …). (2.7)

The structure of V′ as a comodule over P′ is given in 2.15. The elements v_{n+k} for 0 ≤ k ≤ n are permanent cycles. In §2.3 we will consider the effect of inverting v_{n}.

2.2. The Adams-Novikov spectral sequence. Here we have

E_{2}^{s,t} = Ext_{BP_{∗}(BP)}(BP_{∗}, BP_{∗}(X)).

Here we are taking Ext in the category of comodules over the Hopf algebroid BP_{∗}(BP). The difficulty of computing this group is comparable to the classical case.

The structure of BP_{∗}(BP) is as follows. As algebras we have

BP_{∗}(BP) = BP_{∗}[t_{1}, t_{2}, …] with |t_{i}| = 2p^{i} − 2.

It is not a Hopf algebra (i.e., a cogroup object in the category of algebras), but a Hopf algebroid, which is a cogroupoid object in the category of algebras. (For more discussion of this definition see [Rav86, A1.1] or [Rav92a, B.3].) This means that in addition to a coproduct map Δ there is a right unit map η_{R} : BP_{∗} → BP_{∗}(BP). The formulas for these maps involve the formal group law and are somewhat complicated. We will give approximations for them now. Let

I = (p, v_{1}, v_{2}, …) ⊂ BP_{∗}.

Then we have

Δ(t_{i}) ≡ ∑_{j}t_{j} ⊗ t_{i−j}^{p^{j}} mod I where t_{0} = 1
and η_{R}(v_{i}) ≡ ∑_{j}v_{j}t_{i−j}^{p^{j}}  mod I^{2} where v_{0} = p.

There is an analog of (2.30) for Hopf algebroids stated as A1.3.12 in [Rav86]. We have

BP_{∗}(y(n)) = BP_{∗}/I_{n}[t_{1}, …, t_{n}]. (2.8)

The analog of 2.3 is the following.

Corollary 2.9.

Ext_{BP_{∗}(BP)}(BP_{∗}, BP_{∗}(y(n))) = Ext_{BP_{∗}(BP)/(t_{1},... ,t_{n})}(BP_{∗}, BP_{∗}/I_{n}).

When X is a finite complex of type n, the Adams-Novikov E_{2}-term for X̂ is surprisingly easy to compute. In some cases we can get a complete description of it, quite unlike the situation for X itself. It was this computability that originally motivated the second author’s interest in this problem. For such X we know that BP_{∗}(X̂) = BP_{∗}(L_{n}X) = v_{n}^{−1}BP_{∗}(X), and BP_{∗}(X) is always annihilated by some power of the ideal

I_{n} = (p, v_{1}, …, v_{n−1}) ⊂ BP_{∗}.

More generally if X is a connective spectrum in which each element of BP_{∗}(X) is annihilated by some power of I_{n}, we have BP_{∗}(L_{n}X) = v_{n}^{−1}BP_{∗}(X). The results of [Rav87] and the smash product theorem [Rav92a, 7.5.6] imply that the Adams-Novikov spectral sequence for L_{n}X converges to π_{∗}(L_{n}X).

Now assume for simplicity that BP_{∗}(X) is annihilated by I_{n} itself; this condition is satisfied in all of the examples we shall study here. This means that v_{n}^{−1}BP_{∗}(X) is a comodule over v_{n}^{−1}BP_{∗}(BP)/I_{n}, which turns out to be much more manageable than BP_{∗}(BP) itself. There is a change-of-rings isomorphism (originally conceived by Morava and proved in the form we’ll use in [MR77]) that enables us to replace v_{n}^{−1}BP_{∗}(BP)/I_{n} with a smaller Hopf algebra Σ(n), which we now describe. Let

K(n)_{∗} = \{ \begin{matrix}Q & for n = 0 \\ Z/(p)[v_{n}, v_{n}^{−1}] & for n > 0, \end{matrix}

(this is the coefficient ring for Morava K-theory) and define a BP_{∗}- module structure on it by sending v_{m} to zero for m ≠ n . K(n)_{∗} for n > 0 is a graded field in the sense that every graded module over it is free. Then let

Σ(n) = K(n)_{∗} ⊗_{BP_{∗}} BP_{∗}(BP) ⊗_{BP_{∗}} K(n)_{∗},

where the tensor product on the right is with respect to the BP_{∗}-module structure on BP_{∗}(BP) induced by the right unit map η_{R}. Using more precise information about η_{R}, we get the following explicit description of Σ(n) as an algebra.

Σ(n) = K(n)_{∗}[t_{1}, t_{2}, …]/(v_{n}t_{i}^{p^{n}} − v_{n}^{p^{i}}t_{i}).

It is a Hopf algebra with coproduct inherited from that on BP_{∗}(BP). For a BP_{∗}(BP)-comodule M, K(n)_{∗} ⊗_{BP_{∗}} M is a comodule over Σ(n).

Now we can state the change-of-rings theorem of [MR77].

Theorem 2.10. Let M be a BP_{∗}(BP)-comodule that is annihilated by the ideal I_{n}. Then there is a natural isomorphism

Ext_{BP_{∗}(BP)}(BP_{∗}, v_{n}^{−1}M) = Ext_{Σ(n)}(K(n)_{∗}, K(n)_{∗} ⊗_{BP_{∗}} M).

The Ext group on the right is explicitly computable in many interesting cases. It is related to the continuous mod p cohomology of the strict automorphism group of the height n formal group law. This connection was first perceived by Morava and is explained in [Rav86, Chapter 6]. The methods given there lead to the following analog of 2.9.

Corollary 2.11. With BP_{∗}(y(n)) as in (2.8),

Ext_{BP_{∗}(BP)}(BP_{∗}, v_{n}^{−1}BP_{∗}(y(n)))

= Ext_{Σ(n)/(t_{1},... ,t_{n})}(K(n)_{∗}, K(n)_{∗}[v_{n}t_{1}^{p^{n}} − v_{n}^{p}t_{1}, …, v_{n}t_{n}^{p^{n}} − v_{n}^{p^{n}}t_{n}])
= P (v_{n+1}, …, v_{2n}) ⊗ Ext_{Σ(n)/(t_{1},... ,t_{n})}(K(n)_{∗}, K(n)_{∗})
= K(n)_{∗}[v_{n+1}, …, v_{2n}] ⊗ E(h_{n+i,j}: 1 ≤ i ≤ n, 0 ≤ j ≤ n − 1),

where h_{n+i,j} ∈ Ext^{1,2p^{j}(p^{n+i}−1)} corresponds to the primitive element

t_{n+i}^{p^{j}} ∈ Σ(n)/(t_{1}, …, t_{n}).

2.3. The localized Adams spectral sequence. The classical Adams spectral sequence is useless for studying the telescope X̂ because its homology is trivial. We need to replace it with the localized Adams spectral sequence; the first published account of it is due to Miller [Mil81]. It is derived from the Adams spectral sequence in the following way. The telescope X̂ is obtained from X by iterating a v_{n}-map f : X → Σ^{−d}X. Suppose there is a lifting

f̃: X → Σ^{−d}X_{s_{0}}

(where X_{s_{0}} is as in (2.1)) for some s_{0} ≥ 0. This will induce maps f̃: X_{s} → Σ^{−d}X_{s+s_{0}} for s ≥ 0. This enables us to define X̂_{s} to be the homotopy direct limit of

X_{s}  Σ^{−d}X_{s+s_{0}}  Σ^{−2d}X_{s+2s_{0}}  ⋯^{f̃}  w  ^{f̃} w  ^{f̃} w

Let X_{s} = X for s < 0. Thus we get the following diagram, similar to that of (2.1).

⋯  X̂_{−1}  X̂_{0}  X̂_{1}  ⋯ K̂_{−1}  K̂_{0}  K̂_{1},_{_{g_{−1}} u}_{u}  _{_{g_{0}} u}_{u}  _{_{g_{1}} u}_{u}  _{u} (2.12)

where the spectra K̂_{s} are defined after the fact as the obvious cofibers. This leads to a full plane spectral sequence (the localized Adams spectral sequence) with

E_{1}^{s,t} = π_{t−s}(K̂_{s}) and d_{r} : E_{r}^{s,t} → E_{r}^{s+r,t+r−1}

as before. This spectral sequence converges to the homotopy of the homotopy direct limit π_{∗}(lim_{→} X̂_{−s}) if the homotopy inverse limit lim_{←} X̂_{s} is contractible.

Theorem 2.13 (Convergence of the localized Adams spectral sequence). For a spectrum X equipped with maps f and f̃ as above, in the localized Adams spectral sequence for π_{∗}(X̂) we have

• The homotopy direct limit lim_{→} X̂_{−s} is the telescope X̂.

• The homotopy inverse limit lim_{←} X̂_{s} is contractible if the original (unlocalized) Adams spectral sequence has a vanishing line of slope s_{0}/d at E_{r} for some finite r, i.e., if there are constants c and r such that

E_{r}^{s,t} = 0 for s > c + (t − s)(s_{0}/d).

(In this case we say that f has a parallel lifting f̃.)

• If f has a parallel lifting, this localized Adams spectral sequence converges to π_{∗}(X̂).

Proof. For the assertion about the homotopy direct limit, note that

X̂_{s} = lim_{→i} Σ^{−di}X_{s+is_{0}}
so  lim_{→s} X̂_{s} = lim_{→s} lim_{→i} Σ^{−di}X_{s+is_{0}}
= lim_{→i} lim_{→s} Σ^{−di}X_{s+is_{0}}
= lim_{→i} Σ^{−di}X
= X̂.

Next we will prove the assertion about the vanishing line. Let E_{r}^{s,t}(X) denote the E_{r}-term of the Adams spectral sequence for X, and E_{r}^{s,t}(X̂) that of the localized Adams spectral sequence. Then f̃ induces homomorphisms

E_{r}^{s,t}(X) ⟶^{f̃} E_{r}^{s+s_{0},t}(Σ^{−d}X) = E_{r}^{s+s_{0},t+d}(X)

and we have

E_{r}^{s,t}(X̂) = lim_{→k} E_{r}^{s+ks_{0},t+kd}(X),

so the vanishing line of the localized Adams spectral sequence follows from that of the unlocalized Adams spectral sequence.

Next we will show that lim_{←}(X̂_{i}) is contractible. Recall that

X̂_{i} = lim_{→k} Σ^{−kd}X_{i+ks_{0}}

so

π_{m}(X̂_{i}) = lim_{→k} π_{m+kd}(X_{i+ks_{0}}).

Now the vanishing line implies that the map g : X_{s} → X_{s−r+1} satisfies π_{m}(g) = 0 for m < (sd + c)/s_{0}. To see this, note that a permanent cycle of filtration s corresponds to a coset (modulo the image of π_{∗}(X_{s+1})) in π_{∗}(X_{s}). It is dead in the E_{r}-term if and only if its image in π_{∗}(X_{s−r+1}) is trivial.

It follows that for each k > 0 we have a diagram

X_{s}  X_{s−r+1} Σ^{−dk}X_{s+s_{0}k}  Σ^{−dk}X_{s+s_{0}k−r+1}^{g}  w u  u_{^{g} w}

in which both maps g vanish on π_{m} for m < (sd + c)/s_{0}. Hence the map

X̂_{s} ⟶^{ĝ} X̂_{s−r+1}

has the same property.

It follows that if we fix m and s, the homomorphism

π_{m}(X̂_{i}) ⟶ π_{m}(X̂_{s}) (2.14)

is trivial for sufficiently large i, and the image of

lim_{←} π_{∗}(X̂_{i}) ⟶ π_{∗}(X̂_{s})

is trivial for each s, so

lim_{←} π_{∗}(X̂_{i}) = 0.

To complete the proof that lim_{←}(X̂_{i}) is contractible, we need to show that

lim_{←}^{1}π_{∗}(X̂_{i}) = 0.

However, (2.14) implies that the inverse system of homotopy groups is Mittag-Leffler, so lim\underleftarrow{1} vanishes.

According to Boardman [Boa81, §10], the convergence of a whole plane spectral sequence such as ours requires, in addition to the contractibility just proved, the vanishing of a certain obstruction group that he calls W. (It measures the failure of certain direct and inverse limits to commute.) However, his Lemma 10.3 says that our vanishing line implies that W = 0.

Here are some informative examples.

• If we start with the Adams-Novikov spectral sequence, then the map f cannot be lifted since BP_{∗}(f) is nontrivial. Thus we have s_{0} = 0 and the lifting condition requires that X has a horizontal vanishing line in its Adams-Novikov spectral sequence. This is not known (or suspected) to occur for any nontrivial finite X, so we do not get a convergence theorem about the localized Adams-Novikov spectral sequence, which is merely the standard Adams-Novikov spectral sequence applied to X̂.

• If we start with the classical Adams spectral sequence, theorem of Hopkins-Palmieri-Smith says that a type n X (with n > 0) always has a vanishing line of slope 1/|v_{n}| = 1/(2p^{n} − 2), in the E_{r} term, for some r. ([HPS98]) Thus we have convergence if f has a lifting with s_{0} = d/|v_{n}|. This does happen in the few cases where Toda’s complex V (n − 1) exists. Then V (n − 1) is a type n complex with a v_{n}-map with d = |v_{n}| and s_{0} = 1.

• In favorable cases (such as Toda’s examples and y(n)) the E_{2}-term of the localized Adams spectral sequence can be identified as an Ext groups which can be computed explicitly.

We will discuss the last example in more detail. For simplicity we assume until further notice that p is odd. Recall from §2.1 that

Ext_{A_{∗}}(Z/(p), H_{∗}(y(n))) = Ext_{B(n)_{∗}}(Z/(p), Z/(p))

and that the latter can be computed using Cartan-Eilenberg spectral sequence (2.32) for the extension (2.6) with

E_{2} = Ext_{P′_{∗}}(Z/(p), V′).

The effect of localization is to invert v_{n} as in (2.35). The comodule structure on V′ is given by

ψ(v_{2n+i}) = 1 ⊗ v_{2n+i} + ∑_{0≤k<i} ξ_{n+i−k}^{p^{n+k}} ⊗ v_{n+k}
= 1 ⊗ v_{2n+i} + ξ_{n+i}^{p^{n}} ⊗ v_{n} + ….

In the ring v_{n}^{−1}V′, define w_{2n+i} for i > 0 recursively by

w_{2n+i} = v_{n}^{−1} (v_{2n+i} − ∑_{0<k<i} v_{n+k}w_{2n+i−k}^{p^{k}}) , (2.15)

and let

W′ = P (v_{n}, v_{n+1}, ..., v_{2n}, w_{2n+1}, w_{2n+2}, …) ⊂ v_{n}^{−1}V′. (2.16)

(2.15) can be rewritten as

v_{2n+i} = ∑_{0≤k<i−n} v_{n+k}w_{2n+i−k}^{p^{k}}. (2.17)

We will show below (see (5.7)) that for w_{2n+i} as in (2.15),

ψ(w_{2n+i}) = 1 ⊗ w_{2n+i} − \overline{ξ}_{n+i}^{p^{n}} ⊗ 1 + ∑_{n<k<i} \overline{ξ}_{k}^{p^{n}} ⊗ w_{2n+i−k}^{p^{k}}. (2.18)

Then using (2.28) we have

d(w_{2n+i}) = −\overline{ξ}_{n+i}^{p^{n}} ⊗ 1 + ∑_{n<k<i} \overline{ξ}_{k}^{p^{n}} ⊗ w_{2n+i−k}^{p^{k}}.

in the cobar complex C_{P′_{∗}}(W′). Hence the expression on the right is a cocycle, so the same is true of its p^{j−n}th power in the algebra P′_{∗} ⊗ W′, (for j < n, not j ≥ n, as one might expect,)

h_{n+i,j} = −\overline{ξ}_{n+i}^{p^{j}} ⊗ 1 + ∑_{n<k<i} \overline{ξ}_{k}^{p^{j}} ⊗ w_{2n+i−k}^{p^{j+k−n}}, (2.19)

and we denote the element in Ext represented by its transpotent by b_{n+i,j}.

It follows that W′ (2.16) and therefore v_{n}^{−1}V′ are free comodules over

P (ξ_{n+1}^{p^{n}}, ξ_{n+2}^{p^{n}}, …).

Using the localized change-of-rings isomorphism of (2.33), we get

v_{n}^{−1}Ext_{P′_{∗}}(Z/(p), V′)
= Ext_{P′_{∗}}(Z/(p), v_{n}^{−1}V′)
= Ext_{P′_{∗}}(Z/(p), v_{n}^{−1}W′)
= v_{n}^{−1}Ext_{P′_{∗}}(Z/(p), W′)
= v_{n}^{−1}P (v_{n}, …, v_{2n}) ⊗ Ext_{P′_{∗}/(ξ_{n+i}^{p^{n}})}(Z/(p), Z/(p)).

The Ext group above is easy to compute because the coproduct in P′_{∗}/(ξ_{n+i}^{p^{n}}) is trivial, i.e., each generator is primitive. Thus we have

v_{n}^{−1}Ext_{P′_{∗}}(Z/(p), V′)

(2.20) = v_{n}^{−1}P (v_{n}, …, v_{2n}) ⊗ \{ \begin{matrix}E(h_{n+i,j} : i > 0, 0 ≤ j < n) \\ ⊗P (b_{n+i,j} : i > 0, 0 ≤ j < n) \\ for p odd \\ P (h_{n+i,j} : i > 0, 0 ≤ j < n) \\ for p = 2. \end{matrix}

Since the elements v_{n+i} for 0 ≤ i ≤ n are permanent cycles, the Cartan-Eilenberg spectral sequence collapses. There are no multiplicative extensions since E_{∞} has no zero divisors. Hence the above is a description of the E_{2}-term of the localized Adams spectral sequence for Y (n).

2.4. The Thomified Eilenberg-Moore spectral sequence. We will use a Thomified form of the Eilenberg-Moore spectral sequence which is introduced in [MRS].

Let

X ⟶^{i} E ⟶^{h} B (2.21)

be a fiber sequence with simply connected base space B, and suppose that we also have a p-local stable spherical fibration ξ over E which is oriented with respect to mod p homology.

Let Y, and K be the Thomifications of X and E. In [MRS] we construct a diagram

Y  Y_{0}  Y_{1}  Y_{2}  ⋯ K_{0}  K_{1}  K_{2}_{_{g_{0}} u}  _{_{g_{1}} u}_{u}  _{_{g_{2}} u}_{u}  _{u} (2.22)

where Y_{s+1} is the fiber of g_{s} and

H_{∗}(K_{s}) = Σ^{−s}H_{∗}(K) ⊗ \overline{H}_{∗}(B^{(s)}),

for s > 0. This is similar to the Adams diagram of (2.1), but H_{∗}(g_{s}) need not be a monomorphism in general. As before the associated exact couple of homotopy groups leads to a spectral sequence, which we will call the Thomified Eilenberg-Moore spectral sequence.

To identify the E_{2}-term in certain cases, note that H_{∗}(K) is simultaneously a comodule over A_{∗} and (via the Thom isomorphism and the map h_{∗}) H_{∗}(B), which is itself a comodule over A_{∗}. Following Massey-Peterson [MP67], we combine these two structures by defining the semitensor product coalgebra

R_{∗} = H_{∗}(B) ⊗ A_{∗} (2.23) in which the coproduct is the composite

H_{∗}(B) ⊗ A_{∗} H_{∗}(B) ⊗ H_{∗}(B) ⊗ A_{∗} ⊗ A_{∗} H_{∗}(B) ⊗ A_{∗} ⊗ H_{∗}(B) ⊗ A_{∗} ⊗ A_{∗} H_{∗}(B) ⊗ A_{∗} ⊗ A_{∗} ⊗ H_{∗}(B) ⊗ A_{∗} (H_{∗}(B) ⊗ A_{∗}) ⊗ (H_{∗}(B) ⊗ A_{∗}),_{_{Δ_{B}⊗Δ_{A}} u}_{_{H_{∗}(B)⊗ψ_{B}⊗A_{∗}⊗A_{∗}} u}_{_{H_{∗}(B)⊗A_{∗}⊗T ⊗A_{∗}} u}_{_{H_{∗}(B)⊗m_{A}⊗H_{∗}(B)⊗A_{∗}} u} (2.24)

where Δ_{A} and Δ_{B} are the coproducts on A_{∗} and H_{∗}(B), T is the switching map, ψ_{B} : H_{∗}(B) → A_{∗} ⊗ H_{∗}(B) is the comodule structure map, and m_{A} is the multiplication in A_{∗}.

Massey-Peterson gave this definition in cohomological terms. They denoted the semitensor algebra R by H^{∗}(B) ⊙ A, which is additively isomorphic to H^{∗}(B) ⊗ A with multiplication given by

(x_{1} ⊗ a_{1})(x_{2} ⊗ a_{2}) = x_{1}a′_{1}(x_{2}) ⊗ a′′_{1}a_{2},

where x_{i} ∈ H^{∗}(B), a_{i} ∈ A, and a′_{1}⊗a′′_{1} denotes the coproduct expansion of a_{1} given by the Cartan formula. Our definition is the homological reformulation of theirs.

Note that given a map f : V → B and a subspace U ⊂ V, H̄^{∗}(V/U) = H^{∗}(V, U) is an R-module since it is an H^{∗}(V)-module via relative cup products, even if the map f does not extend to the quotient V/U. In our case we have maps G_{s} → B for all s ≥ 0 given by

(e, b_{1}, …, b_{s}) ↦ h_{e},

induced form the standard form of the Eilenberg-Moore spectral sequence, as in [MRS]. These are compatible with all of the maps h_{t}, so H_{∗}(Y_{s}) and H_{∗}(K_{s}) are R_{∗}-comodules, and the maps between them respect this structure.

We will see in the next theorem that under suitable hypotheses, the E_{2}-term of the Thomified Eilenberg-Moore spectral sequence is Ext_{R_{∗}}(Z/(p), H_{∗}(K)) when B is an H-space. When B is an H-space we have a Hopf algebra extension (2.31)

A_{∗} ⟶ R_{∗} ⟶ H_{∗}(B).

This gives us a Cartan-Eilenberg spectral sequence (2.32) converging to this Ext group with

(2.25) E_{2} = Ext_{A_{∗}}(Z/(p), Ext_{H_{∗}(B)}(Z/(p), H_{∗}(K))).

Note that the inner Ext group above is the same as Ext_{H_{∗}(B)}(Z/(p), H_{∗}(E)), the E_{2}-term of the classical Eilenberg-Moore spectral sequence converging to H_{∗}(X). If the latter collapses from E_{2} (which it does in the examples we will study), then the Ext group of (2.25) can be thought of as

Ext_{A_{∗}}(Z/(p), H_{∗}(Y)),

where H_{∗}(Y) is equipped with the Eilenberg-Moore bigrading. This is the usual Adams E_{2}-term for Y when H_{∗}(Y) is concentrated in Eilenberg-Moore degree 0, but not in general.

Theorem 2.26. (i) Suppose that H^{∗}(K) is a free A-module and B is simply connected. Then the Thomified Eilenberg-Moore spectral sequence associated with the homotopy of (2.22) converges to π_{∗}(Y) with

E_{2} = Ext_{R_{∗}}(Z/(p), H_{∗}(K)),

where R_{∗} is the Massey-Peterson coalgebra of (2.23). (ii) If in addition the map i : X → E induces a monomorphism in mod

p homology, then the Thomified Eilenberg-Moore spectral sequence coincides with the classical Adams spectral sequence for Y.

This is proved in [MRS]. Now we give a corollary that indicates that the hypotheses are not as restrictive as they may appear.

Corollary 2.27. Given a fibration

X ⟶ E ⟶ B

with X p-adically complete, a p-local spherical fibration over E, and B simply connected, there is a spectral sequence converging to π_{∗}(Y) (where Y is the Thomification of X) with

E_{2} = Ext_{H_{∗}(B)⊗A_{∗}}(Z/(p), H_{∗}(K)),

where K as usual is the Thomification of E.

Proof. We can apply 2.26 to the product of the given fibration with pt. → Ω^{2}S^{3} → Ω^{2}S^{3}, where Ω^{2}S^{3} is equipped with the p-local spherical fibration of Lemma 3.3 below. Then the Thomified total space is K ∧ H/p, so its cohomology is a free A-module. Thus the E_{2}-term is

Ext_{H_{∗}(B∧H/p)⊗A_{∗}}(Z/(p), H_{∗}(K ∧ H/p)) = Ext_{H_{∗}(B)⊗A_{∗}}(Z/(p), H_{∗}(K)).

2.5. Hopf algebras and localized Ext groups. In this subsection we will collect some results about Ext groups over Hopf algebras and their localizations. We refer the reader to [Rav86, A1.3] for details of the unlocalized theory.

Given a connected graded cocommutative Hopf algebra Γ over a field k (always Z/(p) in this paper) and a left Γ-comodule M, there is a cobar complex C_{Γ}(M) whose cohomology is Ext_{Γ}(k, M); see [Rav86, A1.2.11] where it is denoted by C_{Γ}(k, M). Additively we have

C_{Γ}^{s}(M) = Γ^{⊗s} ⊗ M.

The coboundary on C_{Γ}^{0}(M) = M is given by

d(m) = ψ(m) − 1 ⊗ m (2.28)

where ψ : M → Γ ⊗ M is the comodule structure map. When M is a comodule algebra, C_{Γ}(M) is a differential graded algebra. The product is somewhat complicated and is given in [Rav86, A1.2.15]. For future reference we record the formula for

C_{Γ}^{1}(M) ⊗ C_{Γ}^{1}(M) ⟶^{∪} C_{Γ}^{2}(M),

namely

(γ_{1} ⊗ m_{1}) ∪ (γ_{2} ⊗ m_{2}) = ±γ_{1} ⊗ m′_{1}γ_{2} ⊗ m′′_{1}m_{2}, (2.29)

where m′_{1} ⊗ m′′_{1} denotes the comodule expansion of m_{1}.

Given a Hopf algebra map f : Γ → Φ and a left Φ-comodule M, there is a spectral sequence converging to Ext_{Φ}(k, M) with

(2.29) E_{2}^{i,j} = Ext_{Γ}(k, Ext_{Φ}(k, Γ ⊗ M)) and d_{r} : E_{r}^{s,t} → E_{r}^{s+r,t−r+1}.

It is derived from the double complex C_{Γ}(C_{Φ}(Γ ⊗ M)) by filtering by the first degree. More explicitly we have

C_{Γ}^{i}(C_{Φ}^{j}(Γ ⊗ M)) = Γ^{⊗i} ⊗ Φ^{⊗j} ⊗ Γ ⊗ M

The jth row is C_{Γ}(Φ^{⊗j} ⊗ Γ ⊗ M), which is acyclic since the comodule Φ^{⊗j} ⊗ Γ ⊗ M is free over Γ. This means that filtering by the second degree and computing the cohomology of each row first gives us C_{Φ}(M) in the 0th column. This shows that the total complex is chain homotopy equivalent to C_{Φ}(M) and its cohomology is Ext_{Φ}(k, M).

On the other hand, the ith column is

Γ^{⊗i} ⊗ C_{Φ}(Γ ⊗ M)

so its cohomology is

Γ^{⊗i} ⊗ Ext_{Φ}(k, Γ ⊗ M)

giving

E_{1}^{i,j} = C_{Γ}^{i}(Ext_{Φ}^{j}(k, Γ ⊗ M))

and

E_{2}^{i,j} = Ext_{Γ}^{i}(k, Ext_{Φ}^{j}(k, Γ ⊗ M))

as claimed.

There are two interesting cases of this spectral sequence, occurring when f is surjective and when it is injective. When it is surjective the inner Ext group is Γ□_{Φ}M concentrated in degree 0 since Γ is a free Φ-comodule. Hence the spectral sequence collapses and we have

(2.30) Ext_{Φ}(k, M) = Ext_{Γ}(k, Γ□_{Φ}M).

This is the change-of-rings isomorphism due originally to Milnor-Moore [MM65].

The other interesting case of the spectral sequence occurs when we have an extension of Hopf algebras

Γ ⟶^{f} Φ ⟶^{g} Λ; (2.31)

this means that Φ = Γ ⊗ Λ both as Γ-modules and as Λ-comodules. Applying (2.30) to the surjection g gives

Ext_{Φ}(k, Γ ⊗ M) = Ext_{Φ}(k, Σ□_{Λ}M) = Ext_{Λ}(k, M)

so the E_{2}-term of the spectral sequence associated with f is

(2.32) E_{2}^{i,j} = Ext_{Γ}^{i}(k, Ext_{Λ}^{j}(k, M)).

This is the Cartan-Eilenberg spectral sequence of [CE56, page 349].

Now we will discuss localized Ext groups. Suppose a Hopf algebra Γ has an odd dimensional (this is not needed if k has characteristic 2) primitive element t. Then there is a corresponding element v ∈ Ext_{Γ}^{1}(k, k) which we would like to invert. The class v is represented by a short exact sequence

0 ⟶ k ⟶ L ⟶ Σ^{|t|}k ⟶ 0

of Γ-comodules. Now suppose we have an injective Γ-resolution (such as the one associated with the cobar complex or the double complex above) of a left Γ-comodule M,

0  M  I_{0}  I_{1}  ⋯ w  w  ^{d_{0}} w  ^{d_{1}} w

and let J_{s} = ker d_{s} = cokerd_{s−1}. Then for each s ≥ 0 we have a diagram

0  J_{s}  L ⊗ J_{s}  Σ^{|t|}J_{s}  0 0  J_{s}  I_{s}  J_{s+1}  0_{w}  _{w}  _{w}_{u}  _{w}_{u}_{w}  _{w}  _{w}  _{w}

Using this we get a diagram

M  Σ^{−|t|}J_{1}  Σ^{−2|t|}J_{2}  ⋯ I_{0}  Σ^{−|t|}I_{1}  Σ^{−2|t|}I_{2}  ⋯,_{w}_{u}  _{w}_{u}  _{w}_{u}_{w}  _{w}  _{w}

(where the maps in the bottom row exist because their targets are injective and the vertical maps are inclusions) and hence a direct limit of injective resolutions, of the corresponding cochain complexes obtained by cotensoring over Γ with k, and of Ext groups. We denote the direct limit of cobar complexes by v^{−1}C_{Γ}(M) (the localized cobar complex) and its cohomology by v^{−1}Ext_{Γ}(M), the localized Ext group.

Now suppose we have a map f : Γ → Φ as before with an odd dimensional primitive t ∈ Γ corresponding to v ∈ Ext_{Γ}^{1}(k, M). We can replace the double complex C_{Γ}(C_{Φ}(Γ⊗M)) by v^{−1}C_{Γ}(C_{Φ}(Γ⊗M)). The equivalence between C_{Γ}(C_{Φ}(Γ ⊗ M)) and C_{Φ}(M) is preserved by inverting v in this way, so we get a spectral sequence converging to v^{−1}Ext_{Φ}(k, M). The ith column of the double complex is v^{−1}C_{Γ}^{i}(C_{Σ}(Γ ⊗ M)), and we get

E_{2}^{i,j} = v^{−1}Ext_{Γ}^{i}(k, Ext_{Φ}^{j}(k, M)).

When f is onto, the inner Ext group collapses as before and we get a localized change-of-rings isomorphism

(2.33) v^{−1}Ext_{Φ}(k, M) = v^{−1}Ext_{Γ}(k, Γ□_{Φ}M),

and when f is the injection in a Hopf algebra extension as in (2.31) we get the first form of the localized Cartan-Eilenberg spectral sequence

(2.34) v^{−1}Ext_{Γ}(k, Ext_{Λ}(k, M)) ⟹ v^{−1}Ext_{Φ}(k, M).

We can also consider the case where the odd dimensional primitive t is in Φ but not in Γ. Then we replace the double complex C_{Γ}(C_{Φ}(Γ ⊗ M)) by C_{Γ}(v^{−1}C_{Φ}(Γ ⊗ M)). Then again we have acyclic rows and taking their cohomology gives v^{−1}C_{Φ}(M) in the 0th column. Thus our spectral sequence converges again to v^{−1}Ext_{Φ}(k, M) with

E_{2}^{i,j} = Ext_{Γ}^{i}(k, v^{−1}Ext_{Φ}^{j}(k, M)).

In the case of an extension we use (2.33) to identify the inner Ext group, and we get the second form of the localized Cartan-Eilenberg spectral sequence

(2.35) Ext_{Γ}(k, v^{−1}Ext_{Λ}(k, M)) ⟹ v^{−1}Ext_{Φ}(k, M).

3. The spectra y(n) and Y (n)

We will now construct the spectrum y(n) whose homology and E_{2}- terms were discussed previously, along with the associated telescope Y (n).

3.1. The EHP sequence and some Thom spectra. Recall that ΩS^{3} is homotopy equivalent to a CW-complex with a single cell in every even dimension. Let J_{m}S^{2} (the mth James product of S^{2}) denote its 2m-skeleton. James [Jam55] showed that there is a splitting

ΣΩS^{3} ≃ ⋁_{i>0} S^{2i+1}.

These lead to the James-Hopf maps H_{i}: ΩS^{3} → ΩS^{2i+1} which are surjective in homology. We will denote H_{p} simply by H. When i is a power of a prime p, the p-local fiber of this map is a skeleton, i.e., there is a p-local fiber sequence

J_{p^{n}−1}S^{2} ⟶ ΩS^{3} ⟶ ΩS^{2p^{n}+1}. (3.1)

Definition 3.2. y(n) is the Thom spectrum of the p-local spherical fibration over ΩJ_{p^{n−1}}S^{2} induced from the one over Ω^{2}S^{3} given by Lemma 3.3 below.

y(n) is an A_{∞} ring spectrum, since it is the Thom spectrum of a bundle induced by a loop map ([Mah79].) It may be that in the cases where Toda’s complex V (n − 1) exists and p is odd, that y(n) ≃ V (n − 1) ∧ T (n) (but probably not as A_{∞} ring spectra), where T (n) is the spectrum of [Rav86, §6.5] with

BP_{∗}(T (n)) = BP_{∗}[t_{1}, t_{2}, …, t_{n}].

It is a p-local summand of the Thom spectrum of the canonical complex bundle over ΩSU(p^{n}).

The following is proved in [MRS].

Lemma 3.3. For each prime p there is a p-local spherical fibration over Ω^{2}S^{3} whose Thom spectrum is the mod p Eilenberg-Mac Lane spectrum H/p.

For the rest of this section we assume the p is odd to avoid notational complications. We have

H_{∗}(y(n)) = E(τ_{0}, τ_{1}, ⋯ τ_{n−1}) ⊗ P (ξ_{1}, ⋯ ξ_{n})

as comodules over A_{∗}, as can be inferred from [Mah79].

Lemma 3.4. y(n) is a split ring spectrum, i.e., y(n) ∧ y(n) is a wedge of suspensions of y(n) with one summand for each basis element of H_{∗}(y(n)). In particular

y(n)_{∗}(y(n)) = π_{∗}(y(n)) ⊗ H_{∗}(y(n)).

Proof. Consider the Atiyah-Hirzebruch spectral sequence for y(n)_{∗}(y(n)) with

E_{2} = H_{∗}(y(n); π_{∗}(y(n))).

It suffices to show that each multiplicative generator of H_{∗}(y(n)) is a permanent cycle. These generators all have dimensions no more than |v_{n}|, and below that dimension y(n) is equivalent to H/p. It follows that there are no differentials in the Atiyah-Hirzebruch spectral sequence in that range.

The classical Adams E_{2}-term for y(n) was described in Corollary 2.3. In low dimensions there is no room for any differentials, and we have

Lemma 3.5. Below dimension 2p^{2n+1} − 2p^{n−1} − 2, the Adams spectral sequence for π_{∗}(y(n)) collapses from E_{2} (for formal reasons), with

E_{2} = P (v_{n}, ⋯, v_{2n}) ⊗ E(h_{n+i,j} : i > 0, j ≥ 0) ⊗ P (b_{n+i,j} : i > 0, j ≥ 0), where

v_{n+i} ∈ E_{2}^{1,2p^{n+i}−1}
h_{n+i,j} ∈ E_{2}^{1,2p^{n+i+j}−2p^{j}}
b_{n+i,j} ∈ E_{2}^{2,2p^{n+i+j+1}−2p^{j+1}}.

Proof. From the Hopf algebra extension

H_{∗}(y(n)) ⟶ A_{∗} ⟶ B(n)_{∗}

we see that Ext_{B(n)_{∗}}(Z/(p), Z/(p)) = Ext_{A_{∗}}(Z/(p), H_{∗}y(n)) (our E_{2}- term) is a comodule over H_{∗}(y(n)), regarded as a subalgebra of A_{∗}. From 3.4 we see that this subalgebra of A_{∗} is part of the coalgebra of cooperations in y(n)_{∗}-theory. This means that the corresponding quotient of A acts on the Adams spectral sequence. Routine calculations give

β(v_{n+i}) = h_{n+i,0},
P^{p^{j}}(h_{n+i,j}) = h_{n+i−1,j+1}
and P^{p^{j+1}}(b_{n+i,j}) = b_{n+i−1,j+1}.

Hence if we can show that v_{n+i} for i ≤ n and b_{n+i,0} for i ≤ n − 1 are permanent cycles, then the same will be true of all generators in our range of dimensions. We will show this by proving that there are no elements (besides h_{n+i,0}) in dimension |v_{n+i}| − 1 or |b_{b+i,0}| − 1 for these i.

This can be done by organizing the information in a suitable way. We define the weight ||x|| of an element x by

||v_{n+i}|| = p^{i},
||h_{n+i,j}|| = p^{i+j},
||b_{n+i,j}|| = p^{i+j+1}
and ||xy|| = ||x|| + ||y||.

The generator x having the lowest dimension for its weight is

v_{n}  if ||x|| = 1,
h_{n+1,0} if ||x|| = p and
b_{n+1,j} if ||x|| = p^{j+2} for j ≥ 0.

and the one with the highest weight is always v_{n+i}.

Next observe that for i ≤ n,

(p^{i} + 1)|b_{n+1,j}| > |v_{n+i+j+2}|
and (p^{i} − 1)|v_{n+k}| < |b_{n+1,i+k−2}|.

This means that in our range the target of a differential on a generator x must have the same weight as x.

In general, Adams spectral sequence differentials need not preserve this weight function. However, the first possible exceptions to this occur just outside our range, namely it is possible that

\begin{aligned}d_{1}(v_{2n+1}) & = & v_{n}h_{n+1,n} \\ and d_{2p^{j}}(h_{2n+1−j,j}) & = & v_{n}b_{n+1,n−1−j}^{p^{j}}  for 0 ≤ j ≤ n − 1. \end{aligned} (3.6)

We will see below (5.18) that these differentials actually occur, the first being apparent from the structure of B(n)_{∗}.

Now consider the quantity

μ(x) = |x| − 2p^{n}||x||,

which satisfies μ(xy) = μ(x) + μ(y). Then we have

μ(v_{n+i}) = −2,
μ(h_{n+i,j}) = −1 − 2p^{j}
and μ(b_{n+i,j−1}) = −2 − 2p^{j}.

From this we can see that for any monomial x of weight p^{i}, μ(v_{n+i}) exceeds 1 + μ(x) except when x = h_{n+i,0}, and μ(b_{n+i−1,0}) exceeds it except when x is one of the three generators with a higher value of μ, namely v_{n+i}, h_{n+i,0} and h_{n+i−1,1}.

We know that d_{r}(v_{n+i}) must have weight p^{i} and that μ(d_{r}(v_{n+i})) = μ(v_{n+i})−1, so there is no possible nontrivial value of d_{r}(v_{n+i}). Similarly there can be no differential on b_{n+i−1,0}.

The first positive dimensional element in π_{∗}(y(n)) is v_{n} ∈ π_{2p^{n}−2}(y(n)). We can use the multiplication on y(n) to extend v_{n} to a self-map. The telescope Y (n) is the homotopy colimit of

y(n) ⟶^{v_{n}} Σ^{−|v_{n}|}y(n) ⟶^{v_{n}} Σ^{−2|v_{n}|}y(n) ⟶^{v_{n}} ⋯ (3.7)

Theorem 3.8. The telescope Y (n) defined above is L_{n}^{f}y(n).

Proof. We will adapt the methods used by Hopkins-Smith [HS98] to prove the periodicity theorem, as explained in [Rav92a, Chapter 6]. Let X by a finite complex of type n with a v_{n}-map f such that K(n)_{∗}(f) is multiplication by v_{n}^{k}; see [Rav92a, 6.1.1]. Let R = DX ∧ X, which is a finite ring spectrum. We will compute in π_{∗}(R ∧ y(n)), which is a noncommutative Z/(p)-algebra. Let F ∈ π_{∗}(R ∧ y(n)) denote the image of f under map R → R ∧ y(n), and let G be the image of g = v_{n}^{k} under the map y(n) → R ∧ y(n).

Now R ∧ y(n) has an Adams vanishing line of slope 1/|v_{n}| since y(n) does. The map F −G represents an element above the line of this slope through the origin, so it is nilpotent. (In the proof of the periodicity theorem, the nilpotence theorem was needed at this point. We do not need an analog of it here because we have the vanishing line.) The methods of [Rav92a, 6.1.2] can be applied here to show that for some i > 0, F^{p^{i}} and G^{p^{i}} commute. Now replace F and G by their commuting powers. F − G is still nilpotent for the same reason as before, and for j ≫ 0 we have

0 = (F − G)^{p^{j}} = F^{p^{j}} − G^{p^{j}}.

Thus F^{p^{j}} = G^{p^{j}}. Replacing the original f and g by suitable powers we get a commutative diagram (ignoring suspensions)

X ∧ y(n)  X ∧ y(n) X ∧ y(n)  X ∧ y(n).^{f∧y(n)} w_{_{X∧g} u}  _{u_{X∧g}}_{^{f∧y(n)}w}

It follows that

X̂ ∧ y(n) = X ∧ Y (n) = X̂ ∧ Y (n).

Thus the map y(n) → Y (n) is an X̂_{∗}-equivalence. The result will follow if we can show that Y (n) is X̂_{∗}-local. We have Y (n) ∧ C_{f} = 0, C_{f} being the cofiber of f and therefore a finite complex of type n + 1. Given this, is follows from the thick subcategory theorem that Y (n) annihilates all finite K(n)_{∗}-acyclic complexes, so it is X̂_{∗}-local.

Conjecture 3.9. Y (n) has the same Bousfield class as the telescope associated with a finite complex of type n.

This could be regarded as a new formulation of the telescope conjecture, with Y (n) taking the place of K(n). See also Conjecture 4.4 below. A stronger conjecture is the following.

Conjecture 3.10. Y (n) has the same homotopy type as an infinite wedge of finite type n telescopes.

3.2. The homotopy of L_{n}y(n) and Y (n). We have

BP_{∗}(L_{n}y(n)) = v_{n}^{−1}BP_{∗}(y(n)) = v_{n}^{−1}BP_{∗}/I_{n}[t_{1}, …, t_{n}].

and we know that the Adams-Novikov spectral sequence converges to π_{∗}(L_{n}y(n)). Its E_{2}-term was given above in 2.11, namely

E_{2} = K(n)_{∗}[v_{n+1}, …, v_{2n}] ⊗ E(h_{n+i,j}: 1 ≤ i ≤ n, 0 ≤ j ≤ n − 1).

It follows from 3.5 that each v_{n+i} is a permanent cycle, as is h_{n+i,j} for i + j ≤ n. This accounts for just over half of the n^{2} exterior generators. Perhaps other exterior generators are permanent cycles for dimensional reasons, but we will see below that similar elements in the localized Adams spectral sequence are not.

Question 3.11. Does the Adams-Novikov spectral sequence for L_{n}y(n) collapse?

It does for sparseness reasons when 2p > n^{2}. In any case we have the following result.

Corollary 3.12. π_{∗}(L_{n}y(n)) is finitely presented as a module over the ring

R(n)_{∗} = K(n)_{∗}[v_{n+1}, …, v_{2n}]. (3.13)

We had hoped to show this is not true of π_{∗}(Y (n)) for n > 1, showing that Y (n) (which is L_{n}^{f}y(n)) differs from L_{n}y(n), thereby disproving the telescope conjecture.

We can compute π_{∗}(Y (n)) with the localized Adams spectral sequence. Its E_{2}-term was identified in (2.20) as

E_{2} = R(n)_{∗} ⊗ E(h_{n+i,j} : i > 0, 0 ≤ j ≤ n − 1)
=  ⊗P (b_{n+i,j} : i > 0, 0 ≤ j ≤ n − 1).

As remarked above, the h_{n+i,j} for i+j ≤ n and the v_{n+i} are permanent cycles.

Conjecture 3.14. For i > 0 and 0 ≤ j ≤ n − 1, the element h_{2n+i−j,j} survives to E_{2p^{j}} and supports a nontrivial differential

d_{2p^{j}}(h_{2n+i−j,j}) = v_{n}b_{n+i,n−1−j}^{p^{j}}.

Each b_{n+i,j} for i > 0 and 0 ≤ j ≤ n − 2 survives to E_{1+2p^{n−1}}.

Note that if in addition each b_{n+i,j} were a permanent cycle, then we would have

E_{∞} = R(n)_{∗} ⊗ E(h_{n+i,j}: i + j ≤ n) ⊗ P (b_{n+i,j})/(b_{n+i,j}^{p^{n−1−j}}). (3.15)

For n > 1, this E_{∞} and hence π_{∗}(Y (n)) would be infinitely generated as a module over R(n)_{∗}, which is incompatible with the telescope conjecture. However we cannot prove that each b_{n+i,j} is a permanent cycle for n > 1, and it seems unlikely to be true. Hence we expect E_{∞} to be more complicated than indicated by (3.15).

We have Conjecture 3.16 (Differentials conjecture). In the localized Adams spectral sequence for Y (n) the elements h_{n+i,0} and h_{n+i,1} survive to E_{2} and E_{2p} respectively, and there are differentials

d_{2}(h_{2n+i,0} + s_{2n+i,0}) = v_{n}b_{n+i,n−1}
and  d_{2p}(h_{2n+i−1,1} + s_{2n+i−1,1}) = v_{n}b_{n+i,n−2}^{p}

for decomposables s_{2n+i−j,j}. The elements b_{n+i,j} for j < n − 1 survive to E_{2p+1}, so

E_{2p+1} = R(n)_{∗} ⊗ E(h_{n+i,0} : 1 ≤ i ≤ n) ⊗ E(h_{n+i,1} : 0 ≤ i ≤ n − 1)
⊗ E(h_{n+i,j} : i > 0, 2 ≤ j ≤ n − 1)
⊗ P (b_{n+i,n−2} : i > 0)/(b_{n+i,n−2}^{p})
⊗ P (b_{n+i,j} : i > 0, 0 ≤ j ≤ n − 3).

This will be discussed in §5. The strategy is to lift the computation back to a localized Thomified Eilenberg-Moore spectral sequence converging to Y (n)_{∗}(Ω^{3}S^{1+2p^{n}}) in a manner to be described in §3.3, specifically using the map of (3.18) below. Curiously, its E_{2}-term is essentially the one above tensored with itself. The corresponding statement about differentials there is Conjecture 5.15.

The advantage of this lifting is that the localized Thomified Eilenberg-Moore spectral sequence has far more structure than the localized Adams spectral sequence above, due in large part to the structure of the space Ω^{3}S^{1+2p^{n}}. Its properties are developed in §4. It is an Hspace (which makes the spectral sequence one of Hopf algebras) with a Snaith splitting described in §4.1. The pth Hopf map induces an endomorphism of the spectral sequence that is described in Lemma 5.16.

For n = 1 Conjecture 3.16 gives the following.

E_{2} = R(1)_{∗} ⊗ E(h_{2,0}, h_{3,0}, …) ⊗ P (b_{2,0}, b_{3,0}, …)

with differentials

d_{2}(h_{i+1,0}) = v_{1}b_{i,0} for i ≥ 2,

which leaves

E_{3} = E_{∞} = R(1)_{∗} ⊗ E(h_{2,0}).

Thus for n = 1, the localized Adams spectral sequence and the Adams-Novikov spectral sequence give the same answer. Miller [Mil81] proved the telescope conjecture for n = 1 and p odd by doing a similar calculation with y(1) replaced by V (0).

For n = 2 we have

E_{2} = R(2)_{∗} ⊗ E(h_{3,0}, h_{3,1}, h_{4,0}, h_{4,1}, …) ⊗ P (b_{3,0}, b_{3,1}, b_{4,0}, b_{4,1}, …).

The first differential,

d_{2}(h_{i+2,0}) = v_{2}b_{i,1} for i ≥ 3

gives

E_{3} = R(2)_{∗} ⊗ E(h_{3,0}, h_{4,0}) ⊗ E(h_{3,1}, h_{4,1}, …) ⊗ P (b_{3,0}, b_{4,0}, …).

A pattern of higher differentials consistent with the telescope conjecture would be

d_{1+p^{i+1}}(h_{i+2,1}) = v_{2}^{p^{i+1}}b_{i,0} for i ≥ 3.

Notice that these get arbitrarily long for large i, and they are preempted by the differentials of 3.16,

d_{2p}(h_{i+1,1}) = v_{2}b_{i,0}^{p}.

The splitting of Lemma 3.4 has implications for the spectral sequences we are studying. For any space or spectrum X, y(n)_{∗}(X) is a left comodule over

y(n)_{∗}(y(n)) = H_{∗}(y(n)) ⊗ y(n)_{∗}.

The same goes for E_{r} of a spectral sequence converging to y(n)_{∗}(X), in which case y(n)_{∗}(X) may be filtered in a way compatible with this comodule structure. Similarly Y (n)_{∗}(X) is a left comodule over

Y (n)_{∗}(Y (n)) = H_{∗}(y(n)) ⊗ Y (n)_{∗},

so that any spectral sequence converging to Y (n)_{∗} has a comodule structure over Y (n)_{∗}Y (n).

Lemma 3.17. The comodule structure of the localized E_{2}-term of (2.20) is given by

ψ(v_{n+i}) = ∑_{0≤k≤i} ξ_{i−k}^{p^{n+k}} ⊗ v_{n+k},

ψ(h_{n+i,j}) = ∑_{0≤k≤n−1−j} \overline{ξ}_{k}^{p^{j}} ⊗ h_{n+i−k,j+k}

and  ψ(b_{n+i,j}) = ∑_{0≤k≤n−1−j} \overline{ξ}_{k}^{p^{j+1}} ⊗ b_{n+i−k,j+k}.

Further, in the localized Adams spectral sequence for Y (n), if x is any of the h_{n+i,j} or b_{n+i,j} and d_{2}(x) is nontrivial, it cannot be divisible by v_{n+k} for any k > 0.

Proof. The coalgebra structure in H_{∗}(y(n)) can be read off by injecting it into the dual Steenrod algebra. The values of ψ(v_{n+i}) and ψ(h_{n+i,j}) can be read off from the coproducts on τ_{n+i} and \overline{ξ}_{n+i}^{p^{j}} in A_{∗}, and ψ(b_{n+i,j}) is the transpotent of the latter.

The divisibility of d_{2}(x) by v_{n+k} would contradict this comodule structure.

3.3. The triple loop space. Now we will explain why the triple loop space Ω^{3}S^{1+2p^{n}} is relevant to the proof of the differentials conjecture. Consider the following diagram in which each row is a fiber sequence.

ΩJ_{p^{n}−1}S^{2} × Ω^{3}S^{1+2p^{n}}  Ω_{2}S_{3}  Ω^{2}S^{1+2p^{n}} × Ω^{2}S^{1+2p^{n}} ΩJ_{p^{n}−1}S^{2}  Ω^{2}S^{3}  Ω^{2}S^{1+2p^{n}}_{w}_{u}  _{w}  _{u}_{^{i}  w}  _{w} (3.18)

Here the top row is the Cartesian product of the bottom row with the path fibration on Ω^{2}S^{1+2p^{n}}. The right vertical map is loop space multiplication, while the left one is the product of the identity map on the first factor with the inclusion of the fiber of i on the second factor.

We will look at the Thomified Eilenberg-Moore spectral sequence for each row where the spherical fibration over Ω^{2}S^{3} is the one given by 3.3. Then the bottom row satisfies the hypotheses of Theorem 2.26(ii), so we get the Adams spectral sequence for y(n).

For the top row, the E_{2}-term is described by the following specialization of 2.26.

Theorem 3.19. Consider the Thomified Eilenberg-Moore spectral sequence associated with E = Ω^{2}S^{3}, equipped with the spherical fibration given by 3.3. Suppose the defining fibration has the form

X  E  B X_{1} × ΩB_{2}  Ω^{2}S^{3} × pt.  B_{1} × B_{2}^{i}  w  ^{h}  w_{^{i_{1}×i_{2}} w}  _{^{h_{1}×∗} w}

where h is an H-map and H_{∗}(i_{1}) is monomorphic, and Y = Y_{1} ∧ ΩB_{2+}. Then H_{∗}(Y_{1}) is a subalgebra of A_{∗} = H_{∗}(K), and we let

Γ = A_{∗} ⊗_{H_{∗}(Y_{1})} Z/(p)

Then the E_{2}-term of the Thomified Eilenberg-Moore spectral sequence is

Ext_{H_{∗}(B_{2})⊗Γ}(Z/(p), Z/(p)),

where H_{∗}(B_{2}) ⊗ Γ is a semitensor product coalgebra with coproduct as in (2.23).

In the next section we will see that the top row of (3.18) satisfies the hypotheses of 3.19. In this case the Hopf algebra Γ is B(n)_{∗} of (2.2).

Proof. We have a Hopf algebra extension

H_{∗}(B_{2}) ⊗ A_{∗} ⟶ H_{∗}(B_{1}) ⊗ H_{∗}(B_{2}) ⊗ A_{∗} ⟶ H_{∗}(B_{1})

and hence a Cartan-Eilenberg spectral sequence converging to the E_{2} of the Thomified Eilenberg-Moore spectral sequence with

E_{2} = Ext_{H_{∗}(B_{2})⊗A_{∗}}(Z/(p), Ext_{H_{∗}(B_{1})}(Z/(p), H_{∗}(K))).

In our case H_{∗}(K) is a free comodule over H_{∗}(B_{1}), so the prespectral sequence collapses to

Ext_{H_{∗}(B_{2})⊗A_{∗}}(Z/(p), H_{∗}(Y_{1})). (3.20)

Using the Hopf algebra extension

H_{∗}(B_{2}) ⊗ Γ ⟶ H_{∗}(B_{2}) ⊗ A_{∗} ⟶ H_{∗}(Y_{1})

we can equate (3.20) with

Ext_{H_{∗}(B_{2})⊗Γ}(Z/(p), Z/(p)).

as claimed.

Theorem 3.21. The Thomified Eilenberg-Moore spectral sequence for the top row of (3.18) can be localized in the same way that the one for the bottom row can.

Proof. The spectral sequence in question is based on the diagram (2.22) with Y_{0} = y(n) ∧ Ω^{3}S_{+}^{1+2p^{n}}. This diagram has suitable multiplicative properties. In order to get a localized resolution as in (2.12), we need to lift the map v_{n} ∧ Ω^{3}S_{+}^{1+2p^{n}} to Y_{1}. This lifting exists if and only if g_{0}(v_{n} ∧ Ω^{3}S_{+}^{1+2p^{n}}) is null, which it is since K_{0} = H/p and H_{∗}(v_{n}) = 0.

Thus the Thomified Eilenberg-Moore spectral sequence for the top row of (3.18) can be localized compatibly with our localization of the Adams spectral sequence associated with the bottom row. Convergence of the localization of the top row (which is not actually needed for our purposes) can be proved using the argument of Theorem 2.13 provided we have a suitable vanishing line. Our E_{2}-term is a subquotient of

Ext_{B(n)_{∗}}(Z/(p), Ext_{H_{∗}(Ω^{2}S^{1+2p^{n}})}(Z/(p), Z/(p))).

The connectivities of B(n)_{∗} and Ω^{2}S^{1+2p^{n}} imply that both factors have a vanishing line of slope 1/|v_{n}| as required.

4. Properties of Ω^{3}S^{1+2p^{n}}

4.1. The Snaith splitting. For each n > 0 we have a fibration of spaces (3.1)

J_{p^{n}−1}S^{2} → ΩS^{3} → ΩS^{1+2p^{n}}

which leads to a stable map

Ω^{3}S_{+}^{1+2p^{n}} ⟶^{f} y(n). (4.1)

We know that Ω^{3}S_{+}^{1+2p^{n}} has a Snaith splitting [Sna74]

Ω^{3}S_{+}^{1+2p^{n}} ≃ ⋁_{i≥0} Σ^{|v_{n}|i}T_{i}. (4.2)

Here T_{i} is a certain finite complex (independent of n) with bottom cell in dimension 0 and top cell in dimension 2i−2α(i), where α(i) denotes the sum of the digits in the p-adic expansion of i. In particular T_{1} = S^{0}.

Moreover there are pairings

T_{i} ∧ T_{j} → T_{i+j}.

Thus we get a ring spectrum

T_{∞} = lim_{→} T_{i}. (4.3)

Using the map v_{n} of (3.7) and the map f of (4.1), for each i ≥ 0 we get a diagram

Σ^{i|v_{n}|}T_{i}  Σ^{i|v_{n}|}T_{i+1}  ⋯  Σ^{i|v_{n}|}T_{∞} y(n)  Σ^{−|v_{n}|}y(n)  ⋯  Y (n) w_{_{f} u}  w_{_{f} u}  w _{f̂ u}_{^{v_{n}} w}  _{^{v_{n}} w}  w

The map f on the left is v_{n}^{i} on the bottom cell of its source. The horizontal maps in the top row each multiply the bottom cell by v_{n}, so that each square in the diagram commutes. The map f̂ on the right makes Y (n) a module spectrum over T_{∞}. If 3.9 is true, then the following seems likely.

Conjecture 4.4. The Bousfield class of T_{∞} is that of the p-local sphere spectrum.

T_{∞} is also the Thom spectrum of a bundle over Ω_{0}^{3}S^{3} obtained as follows. From the EHP sequence we get a fiber sequence

Ω^{2}S^{2p−1} ⟶ Ω_{0}^{3}S^{3} ⟶ Ω^{3}S^{2p+1}.

We get our bundle from one over Ω^{3}S^{2p+1} obtained by extending the map S^{2p−2} → BU corresponding to a generator of π_{2p−2}(BU).

Equivalently, our bundle is the one obtained from the map

Ω_{0}^{3}S^{3} = Ω_{0}^{3}SU(2) ⟶ Ω_{0}^{3}SU = BU.

There is also a Hopf map

Ω^{3}S^{1+2p^{n}} ⟶^{H} Ω^{3}S^{1+2p^{n+1}} (4.5)

which is surjective in ordinary homology. It induces a map from the (pi)th Snaith summand of the source to the ith one of the target,

T_{pi} ⟶^{H} Σ^{2(p−1)i}T_{i}, (4.6)

which has degree one on the top cell (in dimension 2pi − 2α(i)). We will use this map to study Ω^{3}S^{1+2p^{n}} and T_{∞} below.

Recall (4.3) that the spectrum T_{∞} is the homotopy direct limit of the Snaith summands of a certain triple loop space. The analogous spectrum obtained from the Snaith splitting of the double loop space of an odd dimensional sphere is H/p, but T_{∞} is far more interesting. It turns out that K(n)_{∗}(T_{∞}) bears a remarkable resemblance to the supposed value of π_{∗}(Y (n)). (Compare Conjecture 3.16 and Theorem 4.17 below.)

4.2. Ordinary homology. H_{∗}(Ω^{3}S^{2dp+1}) has long been known [CLM76] as a module over the Steenrod algebra A, and is as follows.

Lemma 4.7.

H_{∗}(Ω^{3}S^{2dp+1}) = \{ \begin{matrix}P (u_{i} : i ≥ 0) ⊗ P (x_{i,j} : i > 0, j ≥ 0) & for p = 2 \\ P (u_{i} : i ≥ 0) ⊗ E(x_{i,j} : i > 0, j ≥ 0) \\ ⊗ P (y_{i,j} : i > 0, j ≥ 0) & for p > 2 \end{matrix}

where |u_{i}| = 2(p^{i+1}d − 1), |x_{i,j}| = 2p^{j}(p^{i+1}d − 1) − 1 and |y_{i,j}| = 2p^{j+1}(p^{i+1}d − 1) − 2.

For all primes this group can be identified with

Ext_{H_{∗}(Ω^{2}S^{1+2dp})}(Z/(p), Z/(p)),

i.e., the Eilenberg-Moore spectral sequence in mod p homology for the path fibration on Ω^{2}S^{1+2dp} collapses.

For p = 2 the action of the Steenrod algebra A is given by

Sq_{∗}^{2^{k}}(u_{i}) = \{ \begin{matrix}x_{i,0} & if k = 0 \\ u_{i−1}^{2} & if k = 1 \\ x_{i−k+1,k−2}^{2} & otherwise \end{matrix}

Sq_{∗}^{2^{k}}(x_{i,j}) = \{ \begin{matrix}x_{i,j−1}^{2} & if k = 0 and j > 0 \\ x_{i−1,j+1} & if k = j + 1 \\ 0 & otherwise. \end{matrix}

For p odd we have

β_{∗}(u_{i}) = x_{i,0}
P_{∗}^{p^{k}}(u_{i}) = \{ \begin{matrix}−u_{i−1}^{p} & if k = 0 \\ 0 & otherwise \end{matrix}
β_{∗}(x_{i,j}) = y_{i,j−1}  for j > 0
P_{∗}^{p^{k}}(x_{i,j}) = \{ \begin{matrix}x_{i−1,j+1} & if k = j \\ 0 & otherwise \end{matrix}
β_{∗}(y_{i,j}) = 0
P_{∗}^{p^{k}}(y_{i,j}) = \{ \begin{matrix}−y_{i,j−1}^{p} & if k = 0 and j > 0 \\ y_{i−1,j+1} & if k = j + 1 \\ 0 & otherwise. \end{matrix}

We will also need to know the action of the Milnor primitives Q_{k}, which can be read off from Lemma 4.7. Up to sign we have

Q_{k}(u_{i}) = \{ \begin{matrix}x_{i−k,k} & if k < i \\ 0 & otherwise \end{matrix}

(4.8)  Q_{k}(x_{i,j}) = \{ \begin{matrix}y_{i,j−k−1}^{p^{k}} & for 0 ≤ k < j \\ 0 & for k = j \\ y_{i+j−k,k−j−1}^{p^{j}} & for j < k < i + j \end{matrix}

where y_{i,j} = x_{i,j}^{2} when p = 2.

Proof of Lemma 4.7. We will prove this for p odd, leaving the case p = 2 (which is easier) as an exercise for the reader. We will relate the description of the Lemma to the one given by Cohen in [CLM76]. There he speaks of Dyer-Lashof operations with upper indices Q^{s} : H_{q} → H_{q+2(p−1)s}. Within this proof Q_{s} will denote a reindexed Dyer-Lashof operation rather than the Milnor operation. We define Q_{s} : H_{q} → H_{pq+(p−1)s}, when q and s have the same parity, by

Q_{s} = Q^{(s+q)/2}

with

Q_{s}(x) = \{ \begin{matrix}0 & for s < 0 \\ x^{p} & for s = 0. \end{matrix}

With this in mind, Cohen’s result says that

H_{∗}(Ω^{3}S^{2pd+1}) = P (Q_{2}^{i}(u_{0}) : i ≥ 0) ⊗ E(Q_{1}^{j}βQ_{2}^{i}(u_{0}) : i > 0, j ≥ 0)
⊗P (βQ_{1}^{j+1}βQ_{2}^{i}(u_{0}) : i > 0, j ≥ 0),

where u_{0} ∈ H_{2pd−2} is the fundamental class. We define

u_{i} = Q_{2}^{i}(u_{0}),
x_{i,j} = Q_{1}^{j}βQ_{2}^{i}(u_{0})
and y_{i,j} = βQ_{1}^{j+1}βQ_{2}^{i}(u_{0}).

These elements have the indicated dimensions. It remains to show that the action of the Steenrod algebra is as stated.

The action of Steenrod operations on Dyer-Lashof operations is given by the Nishida relations. For operations on a q-dimensional class, these are

P_{∗}^{r}Q_{s} = ∑_{i}(−1)^{r+i}\binom{(p − 1) (\frac{s+q}{2} − r)}{r − pi}Q_{s−2r+2pi}P_{∗}^{i} (4.9)

and

\begin{aligned}P_{∗}^{r}βQ_{s} & = & ∑_{i}(−1)^{r+i}\binom{(p − 1) (\frac{s+q}{2} − r) − 1}{r − pi}βQ_{s−2r+2pi}P_{∗}^{i} \\ & & + ∑_{i}(−1)^{r+i}\binom{(p − 1) (\frac{s+q}{2} − r) − 1}{r − pi − 1}Q_{s+1−2r+2pi}P_{∗}^{i}β. \end{aligned} (4.10)

In particular we have

P_{∗}^{1}Q_{2} = \frac{q}{2}Q_{0},
so P_{∗}^{1}(u_{i}) = P_{∗}^{1}Q_{2}(u_{i−1})
= −Q_{0}(u_{i−1})
= −u_{i−1}^{p}.

For k > 0 (4.9) gives

P_{∗}^{p^{k}}Q_{2} = Q_{2}P_{∗}^{p^{k−1}},
so P_{∗}^{p^{k}}(u_{i}) = P_{∗}^{p^{k}}Q_{2}^{k}(u_{i−k})
= Q_{2}^{k}P_{∗}^{1}(u_{i−k})
= −Q_{2}^{k}(u_{i−k−1}^{p})
= 0 by the Cartan formula.

We have β(u_{i}) = x_{i,0} and β(x_{i,j+1}) = y_{i,j} by definition, and it follows that β(x_{i,0}) = 0 and β(y_{i,j}) = 0.

The Nishida relations also give

P_{∗}^{p^{k}}Q_{1} = \{ \begin{matrix}0 & k = 0 \\ Q_{1}P_{∗}^{p^{k−1}} & k > 0, \end{matrix}

and for s = 1 or 2

P_{∗}^{p^{k}}βQ_{s} = \{ \begin{matrix}−Q_{s−1}β & k = 0 \\ βQ_{s}P_{∗}^{p^{k−1}} & k > 0. \end{matrix}

It follows that

P_{∗}^{1}(x_{i,0}) = P_{∗}^{1}βQ_{2}^{i}(u_{0})
= −Q_{1}βQ_{2}^{i−1}(u_{0})
= −x_{i−1,1},

and for k > 0

P_{∗}^{p^{k}}(x_{i,0}) = P_{∗}^{p^{k}}βQ_{2}^{i}(u_{0})
= \{ \begin{matrix}βQ_{2}^{k}P_{∗}^{1}Q_{2}^{i−k}(u_{0}) & k < i \\ βQ_{2}^{i}P_{∗}^{p^{k−i}}(u_{0}) & k ≥ i \end{matrix}
= 0.

For j > 0,

P_{∗}^{1}(x_{i,j}) = P_{∗}^{1}Q_{1}^{j}βQ_{2}^{i}(u_{0})
= 0,

and when k > 0

P_{∗}^{p^{k}}(x_{i,j}) = P_{∗}^{p^{k}}Q_{1}^{j}(x_{i,0})
= \{ \begin{matrix}0 & k < j \\ Q_{1}^{j}P_{∗}^{p^{k−j}}(x_{i,0}) & k ≥ j \end{matrix}

= \{ \begin{matrix}0 & k < j \\ Q_{1}^{j}(x_{i−1,1}) & k = j \\ 0 & k > j \end{matrix}

= \{ \begin{matrix}x_{i−1,j+1} & k = j \\ 0 & k ≠ j \end{matrix}

as claimed.

Finally we have for j > 0,

P_{∗}^{1}(y_{i,j}) = P_{∗}^{1}βQ_{1}^{j+1}(x_{i,0})
= −Q_{0}βQ_{1}^{j}(x_{i,0})
= −Q_{0}(y_{i,j−1})
= −y_{i,j−1}^{p},

and for all j when k > 0

P_{∗}^{p^{k}}(y_{i,j}) = P_{∗}^{p^{k}}βQ_{1}^{j+1}(x_{i,0})
= βQ_{1}P_{∗}^{p^{k−1}}Q_{1}^{j}(x_{i,0})
= βQ_{1}P_{∗}^{p^{k−1}}(x_{i,j})
= \{ \begin{matrix}0 & k − 1 ≠ j \\ βQ_{1}(x_{i−1,j+1}) & k − 1 = j \end{matrix}
= \{ \begin{matrix}0 & k ≠ j + 1 \\ y_{i−1,j+1} & k = j + 1 \end{matrix}

as claimed.

4.3. Morava K-theory. In this subsection we will study the Eilenberg-Moore spectral sequence for K(n)_{∗}(Ω^{3}S^{2dp+1}) for d > 0. First we need to know K(n)_{∗}(Ω^{2}S^{2dp+1}), which was computed by Yamaguchi [Yam88]. We will assume for simplicity that p is odd. We could find K(n)_{∗}(Ω^{2}S^{2dp+1}) with either the Atiyah-Hirzebruch spectral sequence or the Eilenberg-Moore spectral sequence starting with K(n)_{∗}(ΩS^{2pd+1}). (Since ΩS^{2pd+1} splits after a single suspension, it is easy to work out its Morava K-theory.) It turns out that the two spectral sequences are the same up to reindexing, and we will describe the former.

We have

H_{∗}(Ω^{2}S^{2dp+1}) = E(e_{i} : i ≥ 0) ⊗ P (f_{i} : i ≥ 0) (4.11) with |e_{i}| = 2dp^{i} − 1 and |f_{i}| = 2dp^{i+1} − 2. (When p = 2, f_{i} = e_{i}^{2}.) In terms of the Dyer-Lashof operations Q_{i} we have

e_{i} = Q_{1}^{i}(e_{0})
and f_{i} = βQ_{1}^{i+1}(e_{0})

The coaction of the dual Steenrod algebra A_{∗} is given by

\begin{matrix}e_{i} & ↦ & 1 ⊗ e_{i} + ∑_{0≤k<i} \overline{τ}_{n+k} ⊗ f_{i−n−k}^{p^{n+k}} \\ and f_{i} & ↦ & ∑_{0≤k<i} \overline{ξ}_{k} ⊗ f_{i−k}^{p^{k}}. \end{matrix} (4.12)

In the Atiyah-Hirzebruch spectral sequence there are differentials

d_{1+|v_{n}|}(e_{i+n+1}) = v_{n}f_{i}^{p^{n}} (4.13)

determined by the Milnor operation Q_{n}. This leaves

E_{2p^{n}} = K(n)_{∗} ⊗ E(e_{0}, e_{1}, ⋯ e_{n}) ⊗ P (f_{i})/(f_{i}^{p^{n}}).

Yamaguchi has shown that there are no higher differentials or multiplicative extensions. It is useful to reprove his theorem here.

Theorem 4.14. With notation as above,

K(n)_{∗}(Ω^{2}S^{2dp+1}) = K(n)_{∗} ⊗ E(e_{0}, e_{1}, ⋯ e_{n}) ⊗ P (f_{i})/(f_{i}^{p^{n}}).

Moreover, the elements e_{i} for 0 ≤ i ≤ n and f_{i} for all i ≥ 0 are in the image of k(n)_{∗}(Ω^{2}S^{2dp+1}).

Proof. We need to show that there are no higher differentials or multiplicative extensions. We know that Ω^{2}S^{2dp+1} has a stable splitting which must be respected by all differentials. The Snaith degrees of e_{i} and f_{i−1} are each p^{i}.

We also know that we have a spectral sequence of Hopf algebras, using the Künneth isomorphism for K(n)_{∗}. Hence, differentials must send primitives to primitives. E_{2p^{n}} is primitively generated. The set of primitives with Snaith degree p^{m} is a subset (depending on m) of

{e_{m}, f_{m−1}, f_{m−2}^{p}, ⋯, f_{m−n}^{p^{n−1}}}.

The dimensions of these elements are all within |v_{n}| of each other, so there is no room for higher differentials, and E_{∞} = E_{2p^{n}}.

To show there are no multiplicative extensions, we must show that f_{i}^{p^{n}} = 0 in K(n)_{∗}(Ω^{2}S^{2dp+1}). If this element were nonzero it would be a primitive with Snaith degree p^{i+n+1}. This means it would have to be a linear combination (with coefficients in k(n)_{∗}) of the elements

{f_{i+n}, f_{i+n−1}^{p}, ⋯, f_{i+1}^{p^{n−1}}}.

These elements all have dimensions higher than that of f_{i}^{p^{n}}, so the latter must be zero as claimed.

For the statement about k(n)_{∗}(Ω^{2}S^{2dp+1}), note that in the Atiyah-Hirzebruch spectral sequence for this group, the indicated elements cannot support any nontrivial differentials for the same dimensional reason as before.

In order to use the Eilenberg-Moore spectral sequence (for which the E_{2}-term is Cotor), we need to know the coalgebra structure. This can be described by giving the Verschiebung map V, which is dual to the pth power map. In [Rav93a], the second author proved that

V (v_{n}^{−1}f_{i+n}) = f_{i}^{p^{n−1}}. (4.15)

For the purposes of this calculation, it is convenient to ignore this Verschiebung and proceed as if the Hopf algebra were primitively generated. The resulting Cotor group can be regarded as the E_{1}-term of Tamaki’s spectral sequence, and there will be some d_{1}s reflecting the behavior of V. In other words we can use the Eilenberg-Moore filtration of K(n)_{∗}(Ω^{2}S^{2pd+1}) to set up a spectral sequence converging to the desired Cotor group.

With this in mind, let u_{i} denote the desuspension of e_{i} for 0 ≤ i ≤ n, let x_{i,j} denote that of f_{i}^{p^{j}} for i ≥ 0 and 0 ≤ j < n, and let y_{i,j} be the transpotent of same. In terms of Dyer-Lashof operations on the corresponding classes in ordinary homology, we have

\begin{aligned}u_{i} & = & τ(Q_{1}^{i}(e_{0})) & = & Q_{2}^{i}(u_{0}), \\ x_{i,j} & = & τ(Q_{0}^{j}βQ_{1}^{i}(e_{0})) & = & Q_{1}^{j}βQ_{2}^{i}(u_{0}) \\ and y_{i,j} & = & κ(Q_{0}^{j}βQ_{1}^{i}(e_{0})) & = & βQ_{1}^{j+1}βQ_{2}^{i}(u_{0}) \end{aligned} (4.16)

where τ denotes transgression (or desuspension) and κ denotes Kudo transgression, or transpotence.

Under our indexing conventions the Eilenberg-Moore spectral sequence is in the first quadrant with u_{i}, x_{i,j} ∈ E_{1}^{1,∗} and y_{i,j} ∈ E_{1}^{2,∗}, and d_{r} raises the first index by r. We have

E_{1} = P (u_{k} : 0 ≤ k ≤ n) ⊗ E(x_{i,j} : j < n) ⊗ P (y_{i,j} : j < n).

(In the corresponding ordinary mod p homology Eilenberg-Moore spectral sequence, the E_{2}-term has a similar description, but without upper bounds on the subscripts, and there are no differentials.) It is a spectral sequence of Hopf algebras (again, because of the Künneth isomorphism for K(n)_{∗},) so differentials must send primitives to primitives. Note that the x_{i,j}s above are odd dimensional while the other generators and K(n)_{∗} are even dimensional.

From (4.15) we get

d_{1}(x_{i+n,0}) = v_{n}y_{i,n−1}

(compare this with (4.8) for j = 0 and k = n) so we have

E_{2} = P (u_{i} : 0 ≤ i ≤ n) ⊗ E(x_{i,0} : i ≤ n)
⊗P (y_{i,j} : j ≤ n − 2) ⊗ E(x_{i,j} : 1 ≤ j ≤ n − 1).

As in the case of the Atiyah-Hirzebruch spectral sequence above, we can take advantage of the Snaith splitting and the Hopf algebra structure. We can also take advantage of the Hopf map of (4.5) and (4.6).

Theorem 4.17. In the Eilenberg-Moore spectral sequence for K(n)_{∗}(Ω^{3}S^{2dp+1}), we have

d_{2p^{j}−1}(x_{i+n,j}) = v_{n}y_{i+j,n−1−j}^{p^{j}} for 1 ≤ j ≤ n − 1.

No other differentials occur and E_{∞} = E_{2p^{n−1}}.

For odd primes there are no multiplicative extensions,

K(n)_{∗}(Ω^{3}S^{2dp+1}) = K(n)_{∗}[u_{0}, ⋯ u_{n}] ⊗ E(x_{i,j} : i + j ≤ n)
⊗P (y_{i,j} : j ≤ n − 2)/(y_{i,j}^{p^{n−j−1}}).

Proof. The primitives in E_{2} with Snaith degree p^{m} for m > n are

\begin{matrix}\{x_{m−j,j}: 1 ≤ j ≤ n − 1\} ∪ \\ \{y_{m−ℓ−k−1,k}^{p^{ℓ}}: 0 ≤ k ≤ n − 2, 0 ≤ ℓ + k ≤ m − 2\} ∪ \\ \{u_{k}^{p^{m−k}}: 0 ≤ k ≤ n\}; \end{matrix} (4.18)

for 1 ≤ m ≤ n we have to add the element x_{m,0}.

The dimensions of these elements are

\{ \begin{matrix}|x_{m−j,j}| & = & 2dp^{m} − 2p^{j} − 1, \\ |y_{m−ℓ−k−1,k}^{p^{ℓ}}| & = & 2dp^{m} − 2p^{ℓ+k+1} − 2p^{ℓ} \\ and |u_{k}^{p^{m−k}}| & = & 2dp^{m} − 2p^{m−k} \end{matrix} (4.19)

For m ≤ n, these dimensions are all within |v_{n}| of each other, so there can be no higher differentials in these Snaith degrees. In particular the elements u_{k} for 0 ≤ k ≤ n, x_{i,j} for i + j ≤ n and y_{i,j} for i + j < n are all permanent cycles.

For m = n + 1, (4.18) reads

{x_{n+1−j,j}: 1 ≤ j ≤ n − 1} ∪
{y_{n−j−k,j}^{p^{k}}: 0 ≤ j ≤ n − 2, 0 ≤ j + k ≤ n − 1} ∪
{u_{k}^{p^{n+1−k}}: 0 ≤ k ≤ n}.

The dimensions of these are compatible with the desired differentials

d_{2p^{j}−1}(x_{n+1−j,j}) = v_{n}y_{1,n−1−j}^{p^{j}} for1 ≤ j ≤ n − 1, (4.20)

which can be inferred from (4.8). The only remaining primitives in this Snaith degree are even dimensional, all of the odd dimensional ones having been accounted for.

Now consider the primitives in Snaith degree p^{n+i} for i > 1. We claim that the only differentials that occur here are

d_{2p^{j}−1}(x_{i+n−j,j}) = v_{n}y_{i,n−j−1}^{p^{j}} for j ≤ n − 1. (4.21)

Here we make use of the Hopf map H. Its (i − 1)th iterate is

Ω^{3}S^{1+2p^{n}} ⟶^{H^{i}} Ω^{3}S^{1+2p^{n+i+1}},

which induces a map

T_{pj} ⟶^{H^{i}} Σ^{2(p−1)ij}T_{ij}.

In the Eilenberg-Moore spectral sequence, the induced map sends the source and target of (4.21) to those of (4.20). Thus (4.21) holds provided that there is no earlier differential on x_{i+n−j,j}. Once these differentials have been taken into account, there are no odd dimensional primitives left in Snaith degrees above p^{n}, so the spectral sequence collapses from E_{2p^{n−1}}.

To prove (4.21) note that the elements of (4.19) with ℓ + k < n + 1 have dimensions too high to be a target of a differential on x_{n+i−j,j}, the ones with ℓ + k = n + 1 are the proposed targets, and the one with ℓ + k > n + 1 are powers of elements of lower Snaith degree that have already been killed. The elements u_{k}^{p^{n+i−k}} have either too high a dimension (if i > k) or too high a filtration (if i ≤ k) to be the target of an earlier differential on x_{n+i−j,j}.

It follows that

E_{∞} = E_{2p^{n−1}}
= K(n)_{∗}[u_{0}, ⋯ u_{n}] ⊗ E(x_{i,j} : i + j ≤ n)
⊗P (y_{i,j} : j ≤ n − 2)/(y_{i,j}^{p^{n−j−1}})

This spectral sequence converges to K(n)_{∗}(Ω^{3}S^{2dp+1}).

To show there are no multiplicative extensions, we need to show that y_{i,j}^{p^{n−1−j}} = 0. If it is nonzero, it must be a K(n)_{∗}-linear combination of the elements u_{k}^{p^{n+i−k}} for 0 ≤ k ≤ n, but the latter do not have the right dimensions modulo |v_{n}|, except possibly for p = 2.

4.4. The computation of Y (n)_{∗}(Ω^{3}S^{1+2p^{n}}) via the Eilenberg-Moore spectral sequence. In this subsection we will prove the following.

Theorem 4.22. For each n > 0 there is an additive isomorphism

Y (n)_{∗}(Ω^{3}S^{2dp+1}) = Y (n)_{∗} ⊗_{K(n)_{∗}} K(n)_{∗}(Ω^{3}S^{2dp+1}).

The isomorphism here need not be multiplicative. We will say more about this below after the proof. Before proving his result we need the following.

Lemma 4.23. There is an additive isomorphism

Y (n)_{∗}(Ω^{2}S^{2dp+1}) = Y (n)_{∗} ⊗_{K(n)_{∗}} K(n)_{∗}(Ω^{2}S^{2dp+1}).

Proof. Consider first the Atiyah-Hirzebruch spectral sequence for y(n)_{∗}(Ω^{2}S^{2dp+1}). We have the differentials of (4.13), which leaves

E_{2p^{n}} = y(n)_{∗} ⊗ E(e_{0}, e_{1}, ⋯ e_{n}) ⊗ P (f_{i})/(v_{n}f_{i}^{p^{n}}).

It follows that in the Atiyah-Hirzebruch spectral sequence for Y (n)_{∗}(Ω^{2}S^{2dp+1}) we have

E_{2p^{n}} = Y (n)_{∗} ⊗ E(e_{0}, e_{1}, ⋯ e_{n}) ⊗ P (f_{i})/(f_{i}^{p^{n}}).

Now we can argue as in the proof of 4.14 that there can be no higher differentials for dimensional reasons.

Proof of Theorem 4.22. To prove the theorem we will use Tamaki’s spectral sequence and the computation is essentially the same as that of §4.3. It is not necessary to know π_{∗}(Y (n)) explicitly. In order to use Tamaki’s spectral sequence we will use the computation of Y (n)_{∗}(Ω^{2}S^{2dp+1}) of 4.23. The group Y (n)_{∗}(Ω^{2}S^{2dp+1}) is a free Y (n)_{∗}- module. This means that the E_{2}-term of the Tamaki spectral sequence can be identified as a Cotor group as in the Morava K-theory case.

In this setting, Tamaki’s spectral sequence is one of Hopf algebras, despite the absence of a Künneth isomorphism. The usual structure maps for X an iterated loop space on a sphere give homomophisms both ways between Y (n)_{∗}(X) and Y (n)_{∗}(X × X). Further, we always have a Künneth map Y (n)_{∗}(X) ⊗_{Y (n)_{∗}} Y (n)_{∗}(X) → Y (n)_{∗}(X). In this setting, however, the diagonal map Y (n)_{∗}(X) → Y (n)_{∗}(X × X) need not factor through Y (n)_{∗}(X) ⊗_{Y (n)_{∗}} Y (n)_{∗}(X). We do have a Künneth isomorphism for the E_{2} term of the Tamaki spectral sequence, so that this E_{2} term is a Hopf algebra, and the spectral sequence will be one of Hopf algebras, provided each E_{r} is free over Y (n)_{∗}. The differentials are given below, and none of them introduce any Y (n)_{∗}-torsion into any E_{r}, so we have the desired freeness.

In order to use the Tamaki spectral sequence we need to know the coalgebra structure. As in the computation for Morava K-theory, it is convenient to ignore the nontriviality of the Verschiebung and proceed as if the Hopf algebra were primitively generated. The resulting Cotor group can be regarded as the E_{1}-term of the Tamaki spectral sequence. It is generated as an algebra by elements in cohomological degrees 1 and 2, so the generators must be primitive. With this in mind, we can define u_{i}, x_{i+1,j}, and y_{i+1,j} as in (4.16). As before we have

E_{1} = Y (n)_{∗} ⊗ P (u_{i} : 0 ≤ i ≤ n) ⊗ E(x_{i,0} : i ≤ n)
⊗P (y_{i,j} : j ≤ n − 1) ⊗ E(x_{i,j} : 0 ≤ j ≤ n − 1).

This Eilenberg-Moore spectral sequence maps to the one for Morava K-theory where we have the differentials of (4.21), namely

d_{2p^{j}−1}(x_{i+n−j,j}) = v_{n}y_{i,n−j−1}^{p^{j}} for j ≤ n − 1.

Similar differentials will occur in the spectral sequence at hand if they are not preempted by earlier ones. In Snaith degrees < p^{n+1} we can compare with the Eilenberg-Moore spectral sequence for y(n)_{∗}(Ω^{3}S^{2pd+1}) and conclude that there are no differentials for dimensional reasons, so again the elements u_{i} for i ≤ n, x_{i,j} for i + j ≤ n and y_{i,j} for i + j < n are all permanent cycles.

In Snaith degree p^{n+1}, the primitives are

{x_{n+1−j,j} : 0 ≤ j ≤ n − 1} ∪
{y_{n−j−k,j}^{p^{k}} : 0 ≤ j ≤ n − 1, 0 ≤ j + k ≤ n − 1} ∪
{u_{k}^{p^{n+1−k}} : 0 ≤ k ≤ n}

As in the proof of 4.17,we can use the Hopf map H to rule out many differentials. If some x_{n+1−j,j} supports a nontrivial differential, its target must be in the kernel of H since H(x_{n+1−j,j}) = x_{n−j,j} is a permanent cycle. Thus the target must be a multiple of either y_{1,n−j}^{p^{j}} for 0 ≤ j ≤ n − 1 or u_{0}^{p^{n+1}} . Hence the expected differentials

d_{2p^{j}−1}(x_{n+1−j,j}) = v_{n}y_{1,n−1−j}^{p^{j}}

follow by induction on j.

In larger degrees we can rule out other differentials on x_{n+i,j} as in the proof of 4.17.

In the proof of 4.17 we knew that each y_{i,j} with j ≤ n − 2 was a permanent cycle because the remaining primitives in the same Snaith degree were also even dimensional. However, this argument is not good enough here because we do not know (and we will see that it is not true) that Y (n)_{∗} is even dimensional. We have to consider the possibility that

d_{p^{i+j+1−k}−2}(y_{i,j}) = αu_{k}^{p^{i+j+1−k}} (4.24)

for some α ∈ Y (n)_{∗} and n ≤ k ≤ 2n.

We can use the Hopf map to exclude such a differential in the following way. The Snaith summands of Ω^{3}S^{2dp+1} are independent of d. If d is divisible by p^{m} then our spectral sequence is in the image of the mth iterate of the Hopf map H. Thus we get a diagram

y_{i,j}  αu_{k}^{p^{i+j+1−k}} y_{i+m,j}  α′u_{k′}^{p^{i+j+m+1−k′}}^{d_{r}}  _{w}_{^{d_{r′}} w}_{u_{H^{m}}}  _{u} (4.25)

where

r + 2 = p^{i+j+1−k}
and r′ + 2 = p^{i+j+m+1−k′}

with n ≤, k, k′ ≤ 2n. Since H^{m} induces a map of spectral sequences, we have r ≥ r′, so

i + j + 1 − k ≥ i + j + m + 1 − k′
k′ ≥ k + m

This is incompatible with the upper bound on k′ since we can do this for any value of m, so there can be no nontrivial differential of the form (4.24).

We will now explain why the isomorphism of 4.22 need not be multiplicative. The argument given in the proof of 4.17 to show that there are no multiplicative extensions does not carry over to the computation of Y (n)_{∗}(Ω^{3}S^{1+2p^{n}}). Indeed y_{i,j}^{p^{n−1−j}} could be a nontrivial Y (n)_{∗}-linear combination of the elements u_{k}^{p^{n+i−k}} for 0 ≤ k ≤ n, e.g.,

v_{n}^{−1}b_{i+j+1,n−2−j}u_{0}^{p^{n+i}}

when i + j ≥ n. Here b_{i,j} denotes the image of y_{i−n,j} under the map of (3.18).

The use of the Hopf map in the last paragraph of the proof above is similar to its use by the first author in [Mah77], and it deserves further comment. In that paper there was a 2-local stable splitting

Ω^{2}S^{9} ≃ ⋁_{i>0} Σ^{7i}B_{i}

where the stable summand B_{i} is known now to be the Brown-Gitler spectrum B([i/2]) with bottom cell in dimension 0 and top cell in dimension i − α(i). There one wanted to show that a certain element

x_{j} ∈ Ext_{A_{∗}}^{1,1+2^{j}}(Z/(2), H_{∗}(B_{2^{j}}))

was a permanent cycle. The Hopf map induces

Σ^{−2^{j+1}}B_{2^{j+1}} ⟶^{H} Σ^{−2^{j}}B_{2^{j}}

which sends x_{j+1} to x_{j}. Now suppose there is a nontrivial differential

d_{r}(x_{j}) = z_{j} ∈ Ext_{A_{∗}}^{1+r,r+2^{j}}(Z/(2), H_{∗}(B_{2^{j}})).

It was shown that no such z_{j} is in the image of the Hopf map H, so the differential cannot occur.

This amounts to saying that there is an element

x ∈ π_{0}(lim_{←} Σ^{−2^{j}}B_{2^{j}})

which projects to an element representing x_{j}. Using properties of Brown-Gitler spectra one can produce maps

S^{0} → ΣRP_{−∞}^{−2} → lim_{←} Σ^{−2^{j}}B_{2^{j}}

where the first map is Spanier-Whitehead dual to the transfer map t : RP_{1}^{∞} → S^{0}, and the composite is the desired x.

Dually one can look for an element

x^{∗} ∈ π^{0}(lim_{→} Σ^{2^{j}}DB_{2^{j}})

which is given by the composite

lim_{→} Σ^{2^{j}}DB_{2^{j}} ⟶ RP_{1}^{∞} ⟶^{t} S^{0}.

In [Car83] Carlsson showed that H^{∗}(lim_{→} Σ^{2^{j}}DB_{2^{j}}) has H^{∗}(RP_{1}^{∞}) as a direct summand as an A-module, and that both are unstable injectives. Using results of Goerss-Lannes [GL87] or Lannes-Schwartz [LS89], one can deduce that the spectrum lim_{→} Σ^{2^{j}}DB_{2^{j}} has RP_{1}^{∞} as a retract.

One can ask analogous questions about the triple loop summands T_{i}. The Hopf map induces

Σ^{−2pi}T_{pi} ⟶^{H} Σ^{−2i}T_{i}

for each i. This map sends y_{i+1,j} to y_{i,j}. Our proof shows that for each 0 ≤ j ≤ n − 2 there is an element

y_{j} ∈ lim_{←} Y (n)_{−2p^{j+1}−2}(Σ^{−2p^{k}}T_{p^{k}})

which projects to y_{i,j} for each i. Note that lim_{←} Y (n)_{∗}(Σ^{−2p^{k}}T_{p^{k}}) need not be the same as Y (n)_{∗}(lim_{←} Σ^{−2p^{k}}T_{p^{k}}) since homology does not commute with inverse limits, but it is the former group which interests us here.

The limit problem disappears when we dualize, since homology and cohomology do commute with direct limits. We have an element

y_{j}^{∗} ∈ Y (n)^{2p^{j+1}+2}(lim_{→} Σ^{2p^{k}}DT_{p^{k}}).

and similarly for Morava K-theory.

An analog of Carlsson’s theorem would be the following.

Conjecture 4.26. The spectrum lim_{→} Σ^{2p^{k}}DT_{p^{k}} has a copy of the suspension spectrum of the Eilenberg-Mac Lane space K(Z/(p), 2) as a retract.

Kuhn [Kuh] has recently proved the corresponding statement in cohomology for p = 2.

A proof of 4.26 (or the construction of a suitable map from the direct limit to K(Z/(p), 2)) might lead to an independent construction of the elements y_{j}^{∗} as follows. K(n)^{∗}(K(Z/(p), 2)) is known [RW80] and it is likely that Y (n)^{∗}(K(Z/(p), 2)) has a similar description. The former has n − 1 algebra generators which might map to the desired y_{j}^{∗}.

For more details, see [Rav98]

5. Toward a proof of the differentials conjecture

5.1. The E_{2}-term of the localized Thomified Eilenberg-Moore spectral sequence. Now we are ready to describe our program to prove our conjecture about differentials, 3.16. We will use the map of (3.18) from the localized Thomified Eilenberg-Moore spectral sequence for Y (n)_{∗}(Ω^{3}S^{1+2p^{n}}) to the localized Adams spectral sequence for Y (n). Because the material in this section becomes rather technical at times, we will illustrate each main idea in the case n = 2, where the degree of complexity is quite manageable. To begin this extended example, we recall from Corollary 2.11 that the localized Adams-Novikov spectral sequence for π_{∗}(L_{2}y(2)) has

E_{2} = K(2)_{∗}[v_{3}, v_{4}] ⊗ E(h_{3,0}, h_{4,0}, h_{3,1}, h_{4,1}).

In this case, the localized Adams-Novikov spectral sequence collapses from this point, because the only possible differential (d_{3}(h_{3,1}) = v_{2}^{−p^{4}}h_{3,0}h_{3,1}h_{4,0}h_{4,1}?) is ruled out because h_{3,1} is easily seen to be a permanent cycle in the unlocalized Adams-Novikov spectral sequence for π_{∗}(y(2)).

For all n ≥ 1, we know from §3.3 that the E_{2}-term of the localized Thomified Eilenberg-Moore spectral sequence is

v_{n}^{−1}Ext_{H_{∗}(Ω^{2}S^{1+2p^{n}})⊗B(n)_{∗}}(Z/(p), Z/(p)),

where B(n)_{∗} is as in (2.2) and H_{∗}(Ω^{2}S^{1+2p^{n}}) is given in (4.11).

Lemma 5.1. (i) The E_{2}-term of the localized Thomified Eilenberg-Moore spectral sequence for Y (n)_{∗}(Ω^{3}S^{1+2p^{n}}) is

Ext_{P′} (Z/(p), v_{n}^{−1}Ext_{Q⊗H_{∗}(Ω^{2}S^{1+2p^{n}})}(Z/(p), Z/(p)))
= R(n)_{∗} ⊗ E(h_{n+i,j}) ⊗ P (b_{n+i,j})
⊗P (u_{0}, …, u_{n}) ⊗ E(x̃_{i,j}) ⊗ P (ỹ_{i,j}).

where the indices i and j satisfy i > 0 and 0 ≤ j ≤ n − 1, and

R(n)_{∗} = K(n)_{∗}[v_{v+1}, …, v_{2n}].

The elements x̃_{i,j} and ỹ_{i,j} are related to the homology classes x_{i,j} and y_{i,j} in H_{∗}(Ω^{3}S^{1+2p^{n}}) and will be defined below in (5.9) and (5.10).

(ii) Under the map of (3.18),

v_{n+k} ↦ v_{n+k},
h_{n+i,j} ↦ h_{n+i,j},
b_{n+i,j} ↦ b_{n+i,j},
u_{k} ↦ v_{n+k},
x̃_{i,j} ↦ h_{n+i,j},
and  ỹ_{i,j} ↦ b_{n+i,j}.

(iii) The H_{∗}(y(n))-comodule structure (as in Lemma 3.17) on these generators is given by

ψ(v_{n+i}) = ∑_{0≤k≤i} ξ_{i−k}^{p^{n+k}} ⊗ v_{n+k},

ψ(h_{n+i,j}) = ∑_{0≤k≤n−1−j} \overline{ξ}_{k}^{p^{j}} ⊗ h_{n+i−k,j+k},

ψ(b_{n+i,j}) = ∑_{0≤k≤n−1−j} \overline{ξ}_{k}^{p^{j+1}} ⊗ b_{n+i−k,j+k},
ψ(u_{i}) = 1 ⊗ u_{i} + ∑_{0≤k≤i} \overline{τ}_{k} ⊗ x̃_{i−k,k},

ψ(x̃_{i,j}) = ∑_{0≤k≤n−1−j} \overline{ξ}_{k}^{p^{j}} ⊗ x̃_{i−k,j+k}

and  ψ(ỹ_{i,j}) = ∑_{0≤k≤n−1−j} \overline{ξ}_{k}^{p^{j+1}} ⊗ ỹ_{i−k,j+k}.

In the n = 2 case, the localized Thomified Eilenberg-Moore spectral sequence has

E_{2} = K(2)_{∗}[v_{3}, v_{4}]⊗  E(h_{3,0}, h_{4,0}, …, h_{3,1}, h_{4,1}, …)
⊗P (b_{3,0}, b_{4,0}, …, b_{3,1}, b_{4,1}, …)
⊗P (u_{0}, u_{1}, u_{2}) ⊗ E(x̃_{1,0}, x̃_{2,0}, …, x̃_{1,1}, x̃_{2,1} …)
⊗P (ỹ_{1,0}, ỹ_{2,0}, …, ỹ_{1,1}, ỹ_{2,1} …).

The values of the map of 3.18 and the coaction are easily read off from the Lemma above.

Proof. (i) We begin by describing the coproduct in H_{∗}(Ω^{2}S^{1+2p^{n}}) ⊗ B(n)_{∗} using (2.24) and (4.12). Elements in 1 ⊗ B(n)_{∗} have their usual coproduct, while coproducts for the generators of H_{∗}(Ω^{2}S^{1+2p^{n}})⊗1 are given by

e_{i} ⊗ 1 ↦ e_{i} ⊗ 1 ⊗ 1 ⊗ 1 + 1 ⊗ 1 ⊗ e_{i} ⊗ 1

+ ∑_{0≤k<i−n} 1 ⊗ \overline{τ}_{n+k} ⊗ f_{i−n−k}^{p^{n+k}} ⊗ 1

and f_{i} ⊗ 1 ↦ f_{i} ⊗ 1 ⊗ 1 ⊗ 1 + 1 ⊗ 1 ⊗ f_{i} ⊗ 1

+ ∑_{0<k<i−n} 1 ⊗ \overline{ξ}_{n+k} ⊗ f_{i−n−k}^{p^{n+k}} ⊗ 1.

The best way to get at the cohomology of the semitensor product H_{∗}(Ω^{2}S^{1+2p^{n}}) ⊗ B(n)_{∗} is via the Hopf algebra extension

F ⊗ P′_{∗} ⟶ H_{∗}(Ω^{2}S^{1+2p^{n}}) ⊗ B(n)_{∗} ⟶ E ⊗ Q′_{∗} (5.2)

In the Cartan-Eilenberg spectral sequence for this, we have

E_{2} = Ext_{F ⊗P′_{∗}}(Z/(p), Ext_{E⊗Q′_{∗}}(Z/(p), Z/(p)))
= Ext_{F ⊗P′_{∗}}(Z/(p), U ⊗ V′)

where P′_{∗}, Q′_{∗} and V′ are as in (2.4), (2.5), and (2.7),

U = P (u_{0}, u_{1}, …),
E = E(e_{0}, e_{1}, …),
and F = \{ \begin{matrix}P (f_{1}, f_{2}, …) & for p > 2 \\ P (e_{0}^{2}, e_{1}^{2}, …) & for p = 2. \end{matrix}

The coalgebra structure of F ⊗ P′_{∗} and the structure of U ⊗ V′ as a left comodule over it for odd primes are given by

\{ \begin{matrix}ξ_{i} & ↦ & ∑_{0≤k≤i} ξ_{i−k}^{p^{k}} ⊗ ξ_{k}, \\ f_{i} & ↦ & f_{i} ⊗ 1 + ∑_{0≤k<i} \overline{ξ}_{k} ⊗ f_{i−k}^{p^{k}}, \\ v_{i} & ↦ & ∑_{0≤k≤i} ξ_{i−k}^{p^{k}} ⊗ v_{k} \\ and u_{i} & ↦ & 1 ⊗ u_{i} + ∑_{0≤k<i} \overline{f}_{i−k}^{p^{k}} ⊗ v_{k}, \end{matrix} (5.3)

where ξ_{0} = 1, ξ_{i} = 0 for 0 < i ≤ n, v_{i} = 0 for 0 ≤ i < n, \overline{ξ}_{k} is the conjugate of ξ_{k}, and \overline{f}_{k} is the conjugate of f_{k} in F ⊗ P′_{∗}. These formulas follow from the coproduct in A_{∗} and the coaction of A_{∗} on H_{∗}(Ω^{2}S^{1+2p^{n}}).

As in (2.16) we can enlarge U ⊗ V′ to a ring

Z = P (u_{0}, ..., u_{n}, z_{n+1}, …; v_{n}, ..., v_{2n}, w_{2n+1}, …) ⊂ v_{n}^{−1}U ⊗ V′; (5.4)

where w_{2n+i} is as in (2.15), and z_{n+i} is defined recursively for i > 0 by a similar formula,

z_{n+i} = v_{n}^{−1} (u_{n+i} − ∑_{0<k<i} v_{n+k}z_{n+i−k}^{p^{k}}) . (5.5)

Note that

v_{n}^{(p^{i}−1)/(p−1)}z_{n+i} ∈ U ⊗ V′

and that

v_{n}^{(p^{i}−1)/(p−1)}z_{n+i} ≡ (−1)^{i+1}v_{n+1}^{(p^{i−1}−1)/(p−1)}u_{n+1}^{p^{i−1}} mod (v_{n}). (5.6)

We claim that

ψ(z_{n+i}) = 1 ⊗ z_{n+i} − f_{i}^{p^{n}} ⊗ 1 + ∑_{n<k<i} \overline{ξ}_{k}^{p^{n}} ⊗ z_{n+i−k}^{p^{k}}. (5.7)

The formula for ψ(w_{2n+i}) of (2.18) is the homomorphic image of this under the map of (3.18). We will verify the claim after proving the rest of the lemma.

It follows from (5.7) that Z is free as a comodule over

P (f_{1}^{p^{n}}, f_{2}^{p^{n}}, …) ⊗ P (ξ_{n+1}^{p^{n}}, ξ_{n+2}^{p^{n}}, …)

and that

Ext_{F ⊗P′_{∗}}(Z/(p), Z)

= P (u_{0}, … u_{n}) ⊗ P (v_{n}, …, v_{2n})⊗

Ext_{F/(f_{i}^{p^{n}})⊗P′_{∗}/(ξ_{n+i}^{p^{n}}))}(Z/(p), Z/(p))

= P (u_{0}, … u_{n}) ⊗ P (v_{n}, …, v_{2n})

⊗ \{ \begin{matrix}E(x̃_{i,j}) ⊗ P (ỹ_{i,j}) \\ ⊗E(h_{n+i,j}) ⊗ P (b_{n+i,j}) & for p odd \\ P (x̃_{i,j}) ⊗ P (h_{n+i,j}) & for p = 2, \end{matrix}

(5.8)

where the indices i and j satisfy i > 0 and 0 ≤ j ≤ n − 1, x̃_{i,j} corresponds (roughly speaking; see (5.9) below for the precise definition) to the element −f_{i}^{p^{j}} (which is primitive in the quotient F/(f_{i}^{p^{n}}) ⊗ P′_{∗}/(ξ_{n+i}^{p^{n}})), and ỹ_{i,j} is its transpotent. In particular this Ext group is v_{n}-torsion free.

As in (2.20), localizing inverts v_{n}, and we get the stated value of E_{2} for the Cartan-Eilenberg spectral sequence for (5.2). To show that the spectral sequence collapses from E_{2}, it suffices to show that the elements u_{k} and v_{n+k} for 0 ≤ k ≤ n are permanent cycles. This follows from the fact that e_{k} and τ_{n+k} are primitive for these k.

Next we need to define the elements x̃_{i,j} and ỹ_{i,j}. The localized double complex associated with (5.2) is

C_{F ⊗P′_{∗}}(v_{n}^{−1}C_{E⊗Q′_{∗}}(Z/(p))).

The algebra E ⊗Q′_{∗} is primitively generated, so we can replace its localized cobar complex by its localized Ext group U ⊗v_{n}^{−1}V′. The coaction of F ⊗ P′_{∗} on it was described above in (5.3), and the corresponding local Ext group was given in (5.8). In the cobar complex we have by (5.7) and (2.28),

d(z_{n+i}) = −f_{i}^{p^{n}} ⊗ 1 + ∑_{n<k<i} \overline{ξ}_{k}^{p^{n}} ⊗ z_{n+i−k}^{p^{k}},

so this element is a cocycle. It follows that

x̃_{i,j} = −f_{i}^{p^{j}} ⊗ 1 + ∑_{n<k<i} \overline{ξ}_{k}^{p^{j}} ⊗ z_{n+i−k}^{p^{j+k−n}} (5.9)

is also one.

To define ỹ_{i,j} (the transpotent of x̃_{i,j}), we proceed as follows. We regard x̃_{i,j} as an element in the algebra F ⊗ P′_{∗} ⊗ U ⊗ v_{n}^{−1}V′ and let x̃_{i,j}^{(m)} denote the image of its mth power in the algebra

C_{F ⊗P′_{∗}}^{1}(U ⊗ v_{n}^{−1}V′) = F ⊗ P′_{∗} ⊗ U ⊗ v_{n}^{−1}V′.

Then a routine calculation (using only the fact that x_{i,j} is a cocycle) shows that

d(x̃_{i,j}^{(m)}) = ∑_{0<ℓ<m} \binom{m}{ℓ}x̃_{i,j}^{(ℓ)} ∪ x̃_{i,j}^{(m−ℓ)},

where ∪ denotes the cup product partly described in (2.29). It follows that

\begin{aligned}ỹ_{i,j} & = & ∑_{0<m<p} p^{−1}\binom{p}{m}x̃_{i,j}^{(m)} ∪ x̃_{i,j}^{(p−m)} \\ & = & − ∑_{0<m<p} p^{−1}\binom{p}{m}f_{i}^{p^{j}m} ⊗ f_{i}^{p^{j}(p−m)} ⊗ 1 \\ & & + ∑_{n<k<i} ∑_{0<m<p} p^{−1}\binom{p}{m}\overline{ξ}_{k}^{p^{j+k−n}m} ⊗ \overline{ξ}_{k}^{p^{j+k−n}(p−m)} ⊗ z_{n+i−k}^{p^{1+j+k−n}} + … \end{aligned} (5.10)

is the desired cocycle.

(ii) The assertion about the map of (3.18) follows immediately from the definitions of the elements in question; compare (5.9) with (2.19).

(iii) The first three cases were proved in Lemma 3.17. The last three can be read off from the A_{∗}-comodule structure on H_{∗}(Ω^{3}S^{1+2p^{n}}).

Finally, we need to verify (5.7). We will do so by manipulating the following power series in a dummy variable t. Let

v(t) = ∑_{i≥0} v_{n+i}t^{p^{i}},
v̂(t) = ∑_{i>0} v_{2n+i}t^{p^{n+i}},
\overline{v}(t) = v^{−1}(t),

the functional inverse of v(t),

u(t) = ∑_{i≥0} u_{i}t^{p^{i}},
û(t) = ∑_{i>0} u_{n+i}t^{p^{n+i}},
w(t) = ∑_{i>0} w_{2n+i}t^{p^{n+i}},
z(t) = ∑_{i>0} z_{n+i}t^{p^{n+i}},
ξ(t) = ∑_{i≥0} ξ_{i}^{p^{n}}t^{p^{i}},
\overline{ξ}(t) = ∑_{i≥0} \overline{ξ}_{i}^{p^{n}}t^{p^{i}},
f(t) = ∑_{i≥0} f_{i}^{p^{n}}t^{p^{n+i}} where f_{0} = 1,
and \overline{f}(t) = ∑_{i≥0} \overline{f}_{i}^{p^{n}}t^{p^{n+i}}.

(5.11)

In what follows we will drop the variable t and denote functional composition by the symbol ∘. In this way (5.3) can be rewritten as

ψ(v) = (1 ⊗ v) ∘ (ξ ⊗ 1)
so  ψ(\overline{v}) = (\overline{ξ} ⊗ 1) ∘ (1 ⊗ \overline{v}),
ψ(û) = 1 ⊗ û + (1 ⊗ v) ∘ ((\overline{f} − 1) ⊗ 1),
and  ψ(f − 1) = (\overline{ξ} ⊗ 1) ∘ (1 ⊗ (f − 1)) + (f − 1) ⊗ 1
so  \overline{ξ} ∘ \overline{f} = 1 + \overline{ξ} − f

since in any connected Hopf algebra, ψ(x) = x′⊗x′′ implies that x′x̄′′ = 0.

Similarly (2.15) and (5.5) can be rewritten as

ψ(û) = 1 ⊗ û + (1 ⊗ v) ∘ ((\overline{f} − 1) ⊗ 1),
û = v ∘ z
so  z = \overline{v} ∘ û,
and v̂ = v ∘ w
so w = \overline{v} ∘ v̂.

Thus we have

ψ(z) = ψ(\overline{v}) ∘ ψ(û)
= (\overline{ξ} ⊗ 1) ∘ (1 ⊗ \overline{v}) ∘ (1 ⊗ û + (1 ⊗ v) ∘ ((\overline{f} − 1) ⊗ 1))
= (\overline{ξ} ⊗ 1) ∘ (1 ⊗ \overline{v} ∘ û) + (\overline{ξ} ⊗ 1) ∘ ((\overline{f} − 1) ⊗ 1)
= (\overline{ξ} ⊗ 1) ∘ (1 ⊗ z) + \overline{ξ} ∘ (\overline{f} − 1) ⊗ 1
= (\overline{ξ} ⊗ 1) ∘ (1 ⊗ z) + (1 − f) ⊗ 1,

which is a reformulation of (5.7).

5.2. Short differentials. It follows from 5.1(ii) that a differential on x̃_{i,j} forces a similar one on h_{n+i,j}, and if ỹ_{i,j} is a permanent cycle so is b_{n+i,j}. As in the computations of §4.3 and §4.4 we have more control over such differentials because they must respect the Snaith splitting and the Hopf algebra structure. Thus Conjecture 3.14 is a consequence of the following.

Conjecture 5.12. For i > 0 and 0 ≤ j ≤ n − 1, the element x̃_{n+i−j,j} survives to E_{2p^{j}} and supports a nontrivial differential

d_{2p^{j}}(x̃_{n+i−j,j}) = v_{n}ỹ_{i,n−1−j}^{p^{j}}.

Each ỹ_{i,j} for i > 0 and 0 ≤ j ≤ n − 2 survives to E_{1+2p^{n−1}}.

Note that if in addition each ỹ_{i,j} were a permanent cycle, then we would have

E_{∞} = R(n)_{∗} ⊗ P (u_{0}, …, u_{n}) ⊗ E(x̃_{i,j}, h_{n+i,j}: i + j ≤ n)

⊗ P (ỹ_{i,j}, b_{n+i,j})/(ỹ_{i,j}^{p^{n−1−j}}, b_{n+i,j}^{p^{n−1−j}}). (5.13)

In the n = 2 case, Conjecture 5.12 predicts that all the x̃_{i,0}s survive to E_{2} and that d_{2}(x̃_{3,0}) = v_{2}ỹ_{1,1}, d_{2}(x̃_{4,0}) = v_{2}ỹ_{2,1},, etc. Further, we expect that all the x̃_{i,1}s survive to E_{2p}, and that d_{2p}(x̃_{2,1}) = v_{2}ỹ_{1,0}^{p}, d_{2p}(x̃_{3,1}) = v_{2}ỹ_{2,0}^{p}, etc. If the ỹ_{i,0}s were all permanent cycles, the the localized Thomified Eilenberg-Moore spectral sequence would collapse from E_{2p} and have

E_{∞} = K(2)_{∗}[v_{3}, v_{4}] ⊗P (u_{0}, u_{1}, u_{2}) ⊗ E(h_{3,0}, h_{4,0}, h_{3,1}, x̃_{1,0}, x̃_{2,0}, x̃_{1,1})
⊗P (b_{3,0}, b_{4,0}, …, ỹ_{1,0}, ỹ_{2,0}, …)/(b_{i,0}^{p}, ỹ_{j,0}^{p}).

For all n > 0, one might think that Conjecture 5.12 is a consequence of Theorems 4.17 and 4.22, but this is not the case. The differentials of 4.17 suggest but do not actually imply those of 5.12, because the two spectral sequences are based on different filtrations. The method used in the proof of 4.22 to show that the y_{i,j} are permanent cycles does not imply that the ỹ_{i,j} are.

That method can be used to show that there is an element congruent to ỹ_{i,j} modulo decomposables, namely H^{n+1}(ỹ_{i+n+1,j}) (the image of ỹ_{i+n+1,j} under the (n + 1)th iterate of the Hopf map H), which is a permanent cycle. We can use Lemma 5.16 below to identify it as

H^{n+1}(ỹ_{i+n+1,j}) = ỹ_{i,j} − ∑_{0≤k≤n} β_{i+k,j}(v_{n}^{−1}u_{n−k})^{p^{i+j+k+1−n}} (5.14)

for certain coefficients β_{i+k,j} defined in 5.16. The image of this element under the map of (3.18) is

b_{n+i,j} − ∑_{0≤k≤n} β_{i+k,j}(v_{n}^{−1}v_{2n−k})^{p^{i+j+k+1−n}} = 0 by 5.16 below.

Previously we had thought this image was congruent to b_{n+i,j} modulo decomposables, which would imply the collapsing of the localized Adams spectral sequence for Y (n), but unfortunately this is not the case. Thus the survival of the element of (5.14) is of no help in determining the structure of Y (n)_{∗}. On the other hand, the low dimensional computation showing that x̃_{i,j} survives does imply the survival of h_{n+i,j} for i + j ≤ n.

We have instead

Conjecture 5.15 (Second differentials conjecture). In the localized Adams spectral sequence for Y (n) for n > 1 the elements h_{n+i,0} and h_{n+i,1} survive to E_{2} and E_{2p} respectively, and there are differentials

d_{2}(x̃_{n+i,0} + s_{n+i,0}) = v_{n}ỹ_{i,n−1}
and  d_{2p}(x̃_{n+i−1,1} + s_{n+i−1,1}) = v_{n}ỹ_{i,n−2}^{p}

for decomposables s_{n+i−j,j} (not to be confused with, but mapping to the decomposables s_{2n+i−j,j} of Theorem 3.16). The elements ỹ_{i,j} for j < n − 1 survive to E_{2p+1}, so

E_{2p+1} = R(n)_{∗} ⊗ P (u_{0}, …, u_{n}) ⊗ E(x̃_{i,0}, h_{n+i,0} : 1 ≤ i ≤ n)
⊗ E(x̃_{i,1}, h_{n+i,1} : 0 ≤ i ≤ n − 1)
⊗ E(x̃_{i,j}, h_{n+i,j} : i > 0, 2 ≤ j ≤ n − 1)
⊗ P (ỹ_{i,n−2}, b_{n+i,n−2} : i > 0)/(ỹ_{i,n−2}^{p}, b_{n+i,n−2}^{p})
⊗ P (ỹ_{i,j}, b_{n+i,j} : i > 0, 0 ≤ j ≤ n − 3).

We will make use of the Hopf map as before. Consider the following diagram in which both rows are fiber sequences.

ΩJ_{p^{n}−1}S^{2} × Ω^{3}S^{1+2p^{n}}  Ω^{2}S^{3} × pt.  Ω^{2}S^{1+2p^{n}} × Ω^{2}S^{1+2p^{n}} ΩJ_{p^{n}−1}S^{2} × Ω^{3}S^{1+2p^{m+n}}  Ω^{2}S^{3} × pt.  Ω^{2}S^{1+2p^{n}} × Ω^{2}S^{1+2p^{m+n}} w_{_{ΩJ_{p^{n}−1}S^{2}×H^{m}} u}  w  _{u_{Ω^{2}S^{1+2p^{n}}×H^{m}}} w  w

This gives us a map from the Thomified Eilenberg-Moore spectral sequence for y(n)_{∗}(Ω^{3}S^{1+2p^{n}}) to the one for y(n)_{∗}(Ω^{3}S^{1+2p^{n+k}}). Now the second triple loop space has the same Snaith summands as the first, so the two spectral sequences are isomorphic. Thus the Hopf map H induces an endomorphism of our spectral sequence which is multiplicative and linear over R(n)_{∗} ⊗ E(h_{n+i,j}) ⊗ P (b_{n+i,j}).

Lemma 5.16. The Hopf map described above sends

u_{k+1} ↦ u_{k},
x̃_{i+1,j} ↦ x̃_{i,j} − η_{i,j}(v_{n}^{−1}u_{n})^{p^{i+j−n}}
and  ỹ_{i+1,j} ↦ ỹ_{i,j} − β_{i,j}(v_{n}^{−1}u_{n})^{p^{i+j+1−n}}.

where the coefficients η_{i,j} and β_{i,j} vanish for i ≤ n and are defined recursively by

h_{n+i,j} = ∑_{0≤k≤n} η_{i+k,j}(v_{n}^{−1}v_{2n−k})^{p^{i+j+k−n}}
≡ η_{n+i,j} mod (v_{n+1}, …, v_{2n})
and  b_{n+i,j} = ∑_{0≤k≤n} β_{i+k,j}(v_{n}^{−1}v_{2n−k})^{p^{i+j+k+1−n}}
≡ β_{n+i,j} mod (v_{n+1}, …, v_{2n}).

This along with Conjecture 3.16 implies that

d_{2p^{j}}(η_{n+i−j,j}) = v_{n}β_{i,n−1−j}^{p^{j}} (5.17)

for j = 0, 1.

Proof. The value on u_{k+1} is immediate.

To evaluate H(x̃_{i+1,j}) we need first to compute H(z_{n+i+1}) for z_{n+i+1} as defined in (5.5). Let

r_{i} = \{ \begin{matrix}0 & for i = 0 \\ v_{n}^{−1} (v_{n+i} − ∑_{0<k<i} v_{n+k}r_{i−k}^{p^{k}}) & for i > 0. \end{matrix}

We will show by induction on i that

H(z_{n+i+1}) = \{ \begin{matrix}v_{n}^{−1}u_{n} & for i = 0 \\ z_{n+i} − (v_{n}^{−1}u_{n})^{p^{i}}r_{i} & for i > 0. \end{matrix}

This is immediate for i = 0. For the inductive step with i > 0, write

z_{n+i+1} = v_{n}^{−1} (u_{n+i+1} − ∑_{0<k≤i} v_{n+k}z_{n+i+1−k}^{p^{k}}) ,

so we have

H(z_{n+i+1}) = v_{n}^{−1} (H(u_{n+i+1}) − ∑_{0<k≤i} v_{n+k}H(z_{n+i+1−k}^{p^{k}}))

= v_{n}^{−1} (u_{n+i} − ∑_{0<k≤i} v_{n+k}(z_{n+i−k} − r_{i−k}(v_{n}^{−1}u_{n})^{p^{i−k}})^{p^{k}})
= z_{n+i} − v_{n}^{−1}v_{n+i}(v_{n}^{−1}u_{n})^{p^{i}} + ∑_{0<k≤i} v_{n+k}r_{i−k}^{p^{k}}(v_{n}^{−1}u_{n})^{p^{i}}
= z_{n+i} − r_{i}(v_{n}^{−1}u_{n})^{p^{i}}

as claimed.

Now the Hopf map commutes with the coboundary, so we have

H(x̃_{i+1,n}) = H(d(z_{n+i+1}))
= d(H(z_{n+i+1}))
= d(z_{n+i} − w_{n+i}(v_{n}^{−1}u_{n})^{p^{i}})
= x̃_{i,n} − d(w_{n+i})(v_{n}^{−1}u_{n})^{p^{i}}.

It follows that H(x̃_{i,j}) is as indicated where

η_{i,n} = d(w_{n+i})

and η_{i,j} is its p^{j−n}th power. For i ≤ n this vanishes since w_{n+i} is a cocycle.

For the recursive definition we use the series notation of (5.11) with

r(t) = ∑_{i>0} r_{i}t^{p^{i}},

so we have

v ∘ r = v − v_{n}.

It follows that

r = \overline{v} ∘ (v − v_{n})
= 1 − \overline{v} ∘ v_{n}
so r ∘ v_{n}^{−1} ∘ (v − v̂) = v_{n}^{−1} ∘ (v − v̂) − \overline{v} ∘ (v − v̂)
= v_{n}^{−1} ∘ (v − v̂) − 1 + \overline{v} ∘ v̂
= v_{n}^{−1} ∘ (v − v̂) − 1 + w

Now the expression v_{n}^{−1} ∘ (v − v̂) is concentrated in dimensions below that of w_{2n+1}. Thus for each i > 0 we have

w_{2n+i} = ∑_{0≤k≤n} r_{i+k}(v_{n}^{−1}v_{2n−k})^{p^{i+k}}.

Taking the coboundary gives

h_{n+i,n} = ∑_{0≤k≤n} η_{i+k,n}(v_{n}^{−1}v_{2n−k})^{p^{i+k}}.

so

h_{n+i,j} = ∑_{0≤k≤n} η_{i+k,j}(v_{n}^{−1}v_{2n−k})^{p^{i+j+k−n}}.

Taking the transpotent of the above gives the desired formula for H(ỹ_{i+1,j}).

The proof of 5.1(i) can be modified to give a similar description of the unlocalized E_{2}-term in Snaith degrees less than p^{n+1}. One can make an argument similar to that of 3.5 to show that there are no differentials in that range.

Alternately, one can look at the ordinary Adams spectral sequence for y(n)_{∗}(Ω^{3}S^{1+2p^{n}}). Using the skeletal filtration one gets a prespectral sequence converging to the Adams E_{2}-term with

E_{2} = H_{∗}(Ω^{3}S^{1+2p^{n}}) ⊗ Ext_{B(n)_{∗}}(Z/(p), Z/(p)).

Again there is no room for differentials in the range of the homology of the p^{n}th Snaith summand. The first differentials occur in Snaith degree p^{n+1}. They are induced by the Milnor operation Q_{n} given in (4.8), namely

\begin{aligned}d_{2p^{n}−1}(u_{n+1}) & = & v_{n}x_{1,n} and \\ d_{2p^{n}−1}(x_{n+1−j,j}) & = & v_{n}y_{1,n−1−j}^{p^{j}} for 0 ≤ j ≤ n − 1, \end{aligned} (5.18)

where differentials are indexed by the skeletal filtration. These give the differentials of Conjecture 5.15 for i = 1, and (via the map of (3.18)) those of (3.6).

This enables us to proceed by induction on Snaith degree, using the Hopf endomorphism described in Lemma 5.16. Assume inductively that the differentials on x̃_{n+i−j−1,j} and η_{n+i−j−1,j} are as stated in 3.16 and (5.17). Differentials must commute with the Hopf map, so if x̃_{n+i−j,j} survives to E_{2p^{j}} we have

H(d_{2p^{j}}(x̃_{n+i−j,j})) = d_{2p^{j}}(H(x̃_{n+i−j,j}))
= d_{2p^{j}}(x̃_{n+i−j−1,j} − η_{n+i−j−1,j}(v_{n}^{−1}u_{n})^{p^{i−1}}))
= v_{n}y_{i−1,n−1−j}^{p^{j}} − v_{n}β_{i−1,n−1−j}^{p^{j}}(v_{n}^{−1}u_{n})^{p^{i−1}}
= H(v_{n}ỹ_{i,n−1−j}^{p^{j}}).

This means that if x̃_{n+i−j,j} survives to E_{2p^{j}} then

d_{2p^{j}}(x̃_{n+i−j,j}) = v_{n}ỹ_{i,n−1−j}^{p^{j}} + c_{i,j}, (5.19)

where the error term c_{i,j} must be a Hopf algebra primitive of Snaith degree p^{n+i} that is in the kernel of the Hopf endomorphism H.

The other possibility is that x̃_{n+i−j,j} does not survive to E_{2p^{j}} but supports an earlier differential of the form

d_{r}(x̃_{n+i−j,j}) = c_{i,j} for 2 ≤ r < 2p^{j}, (5.20)

where c_{i,j} is as above. Similarly we can assume inductively that ỹ_{n+i−j,j−1} survives to E_{2p+1}, so a differential on ỹ_{n+i−j,j−1} must have the form

d_{r}(ỹ_{n+i−j,j−1}) = c′_{i,j} for 2 ≤ r ≤ 2p, (5.21)

where c′_{i,j} has the same properties as c_{i,j}. We will refer to such unwanted differentials as spurious and show they cannot occur by showing that there are no nontrivial elements c_{i,j} and c′_{i,j} as above. We will use the structure of the triple loop space Ω^{3}S^{1+2p^{n}}.

5.3. Excluding spurious differentials. The fact that the error terms c_{i,j} and c′_{i,j} are primitives of Snaith degree p^{n+i} in the kernel of the Hopf endomorphism means that they must have the form

(5.22)  c_{i,j} = γ_{i,j}u_{0}^{p^{n+i}} + ∑_{0≤k≤n−1} α_{i,j,k}ỹ_{1,k}^{p^{n+i−k−2}}
(5.23)  and c′_{i,j} = γ′_{i,j}u_{0}^{p^{n+i}} + ∑_{0≤k≤n−1} α′_{i,j,k}ỹ_{1,k}^{p^{n+i−k−2}}

where the coefficients γ_{i,j},α_{i,j,k}, γ′_{i,j} and α′_{i,j,k} are in the E_{r}-term of the localized Adams spectral sequence for Y (n) as follows.

γ_{i,j} ∈ E_{r}^{1+r−p^{n+i}, 3p^{n+i}−2p^{j}+r−2}
α_{i,j,k} ∈ E_{r}^{1+r−2p^{n+i−k−2}, 2p^{n+i−1}−2p^{j}+r−1}
γ′_{i,j} ∈ E_{r}^{2+r−p^{n+i}, 3p^{n+i}−2p^{j}+r−2}
α′_{i,j,k} ∈ E_{r}^{2+r−2p^{n+i−k−2}, 2p^{n+i−1}−2p^{j}+r−1}

Note that there is no hope of excluding these coefficients by simple sparseness arguments, because there are only finitely positive values of t for which the group E_{r}^{s, s|v_{n}|+t} vanishes for small r.

These filtrations of these coefficients are negative, but the spurious differentials must lift back to the unlocalized Thomified Eilenberg-Moore spectral sequence, and there the coefficients must have nonnegative filtration. The element x̃_{i,j} or ỹ_{i,j} need not be in the image of the unlocalized E_{r}-term, but some v_{n}-multiple of each must be until we get to the stage where it supports a localized differential. The power of v_{n} could increase with r if there is an unlocalized d_{r} with a target in the v_{n}-torsion.

Thus in order to exclude spurious differentials, we will proceed as follows.

(i) Find the smallest v_{n}-multiple of x̃_{i,j} which is in the image of the unlocalized E_{2}-term.

(ii) Get an upper bound (depending on dimension) of the v_{n}-torsion

in the unlocalized E_{2}-term. (iii) Show that the torsion created in E_{3} by the expected d_{2}s does not

exceed this upper bound. (iv) Use the torsion estimate to get information about smallest v_{n}-

multiple of x̃_{i,j} which is in the image of the unlocalized E_{r}-term. This will lead to restrictions on the coefficients in (5.22) and (5.23) which will enable us to exclude spurious differentials.

We do not know how to control the torsion in E_{2p+1} created by the d_{2p}s, and this difficulty prevents us from proving Conjectures 3.14 and 5.12 for j > 1.

For step (i) above, let

e(i, j) = \{ \begin{matrix}0 & for i ≤ 0 \\ \frac{p^{i+j}−p^{j}}{p−1} & for i > 1. \end{matrix} (5.24)

Then we can combine (5.6) with (5.9) to conclude that for i > 1

v_{n}^{e(i−1,j+1)}x̃_{n+i,j} ≡ (−1)^{i}v_{n+1}^{e(i−2,j+1)}\overline{ξ}_{n+1}^{p^{j}} ⊗ u_{n+1}^{p^{i+j−1}} mod (v_{n}). (5.25)

This represents (−1)^{i}v_{n+1}^{e(i−2,j+1)}h_{n+1,j}u_{n+1}^{p^{i+j−1}}, which is nontrivial in the appropriate Ext group. Similarly one can show that v_{n}^{e(i−1,j+2)}ỹ_{n+i,j} has a nontrivial reduction modulo v_{n}.

For step (ii) above we have the following torsion estimate.

Lemma 5.26. All v_{n}-torsion in the E_{2} of the Thomified Eilenberg-Moore spectral sequence below dimension 2p^{n}(p^{n+i} + p^{n+1} − 2) is killed by v_{n}^{e(i,0)}.

Proof. Recall from the proof of 5.1 that our E_{2}-term is isomorphic (up to regrading) to

Ext_{F ⊗P′_{∗}}(Z/(p), U ⊗ V′).

Consider the short exact sequence of comodules over F ⊗ P′_{∗},

0 → U ⊗ V′ → Z → Z/(U ⊗ V′) → 0,

where Z is an in (5.4). We know by (5.8) that the Ext group for Z is torsion free, so the torsion in E_{2} all comes from the Ext group for the quotient comodule via the connecting homomorphism. It follows that the torsion in E_{2} is controlled by that in the quotient itself. The first element there not killed by v_{n}^{e(i,0)} is z_{n+1}z_{n+i}, which is in the indicated dimension.

For (iii), the localized differential d_{2}(x̃_{n+i,0}) = v_{n}ỹ_{i,n−1} pulls back to

d_{2}(v_{n}^{e(i−1,1)}x̃_{n+i,0}) = v_{n}^{e(i,0)}ỹ_{i,n−1}
= \{ \begin{matrix}v_{n}^{e(i,0)}ỹ_{i,n−1} & for i ≤ n + 1 \\ v_{n}^{e(n+1,0)}(v_{n}^{e(i−n−1,n+1)}y_{i,n−1}) & for i ≥ n + 2, \end{matrix}

so in E_{3} the element v_{n}^{e(i−n−1,n+1)}y_{i,n−1} for i ≥ n + 2, which is not divisible by v_{n}, has dimension

\frac{2p^{n}(p^{n+1} − 1)(p^{i} − 1)}{p − 1} − 2,

and is killed by v_{n}^{e(n+1,0)}. This exponent does not exceed the one given by 5.26, so no higher torsion exists in E_{3}.

We now turn to step (iv). In (5.22) and (5.23) we can ignore the terms with k = n − 1 since we know that ỹ_{1,n−1} is killed by d_{2}. The remaining coefficients with the largest filtrations are α_{i,j,n−2} and α′_{i,j,n−2} with

Filt(α_{i,j,n−2}) = 1 + r − 2p^{i}
Filt(α′_{i,j,n−2}) = 2 + r − 2p^{i}

Thus in order to get a spurious value of d_{2}(v_{n}^{e(i−1,1)}x̃_{n+i,0}) we would need the quantity

3 + e(i − 1, 1) − 2p^{i} = (3 − 2p)e(i, 0)

to be positive, but it never is. Thus d_{2}(x̃_{n+i,0}) is as claimed.

For the differential on x̃_{n+i−1,1}, we need to estimate the smallest v_{n}- multiple of it which is in the image of the unlocalized E_{r}-term. If we assume the worst, namely that at each stage there is a differential with a target having the largest order of v_{n}-torsion allowed by 5.26, namely e(i, 0). Then the filtration of this element is at most

1 + e(i − 1, 1) + (r − 2)e(i, 0) = (r − 1)e(i, 0). (5.27)

It follows that for r < 2p the filtration of v_{n}^{e(i−1,1)+(r−2)e(i,0)}α_{i,1,n−2} is at most 2p − 3. The product of any other coefficient with this power of v_{n} would have negative filtration, so the other coefficients must vanish.

Now we make use of the comodule structure of Lemma 5.1(iii). It implies that α_{i,1,n−2} cannot be divisible by v_{n+k} for k > 0. This means it suffices to consider it modulo (v_{n+1}, …, v_{2n}). Its image there is a linear combination of elements of the form v_{n}^{e}x with e ≥ 0 and x in the subring generated by the surviving h_{n+i,j} and b_{n+i,j}. The filtration of x must be a nonnegative multiple of 2p − 2, so it must be 0. This means our spurious differential has the form

d_{r}(x̃_{n+i−1,1}) = v_{n}^{e}ỹ_{1,n−2}^{p^{i}}.

The exponent e is positive, which contradicts our assumption that r < 2p.

However we cannot exclude the case r = 2p, so in the localized Thomified Eilenberg-Moore spectral sequence for some i ≥ 3 we could have

d_{2p}(x̃_{n+i−1,1}) = v_{n}ỹ_{n+i−2,0}^{p} + α_{n−2}y_{1,n−2}^{p^{i}}
so d_{2p}(x̃_{n+i−1,1} − v_{n}^{−1}α_{n−2}y_{1,n−2}^{p^{i}−p}x̃_{n,1}) = v_{n}ỹ_{n+i−2,0}^{p}.

We still need to show that the elements ỹ_{n+i−j,j−1} for 0 < j < n and x̃_{n+i−j,j} for 1 < j < n survive to E_{2p+1}. An argument similar to the one above shows that each survives to E_{2p}. At that stage we know that ỹ_{1,n−2}^{p} gets killed, so we can ignore the coefficients α_{i,j,n−2} and α′_{i,j,n−2}, concentrating instead on α_{i,j,n−3} and α′_{i,j,n−3}. Multiplying them by the worst power of v_{n} still gives an element with negative filtration, so we can exclude spurious d_{2p}s on these elements.

This completes the program to prove Conjectures 3.16 and 5.15.

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