The Fiber of Iterated Freudenthal Suspension and Morava K-theory of Ω^{k}S^{2ℓ+1}

Dai Tamaki Department of Mathematical Sciences Faculty of Science Shinshu University Matsumoto, 390-8621 Japan

January 29, 2001

Abstract We propose a new approach to Morava K-theory of iterated loop space on a sphere. The fiber of iterated Freudenthal suspension map plays a central role. In order to make use of it, we construct two kinds of strong convergent cobar-type Eilenberg-Moore spectral sequence related to the fiber of iterated Freudenthal suspension. The little cube model for the fiber of iterated Freudenthal suspension found by Wong in 1994 is used. As a test case of our approach, K(n)_{∗}(Ω^{4}S^{2ℓ+1}) is determined.

1 Introduction

The fiber W (ℓ) of the double suspension map on an odd dimensional sphere

E^{2} : S^{2ℓ−1} ⟶ Ω^{2}S^{2ℓ+1}

has been playing essential roles in the study of the global structure of the unstable homotopy groups of spheres, including the v_{1}-periodic homotopy by Mahowald and Thompson [7, 15]. Localized at a prime p, W (ℓ) is known to be equipped with a map

σ : W (ℓ) ⟶ Ω^{2p}W (ℓ + 1)

which is degree one on the bottom cell [2]. This map, so-called the secondary suspension map, induces a v_{1}-periodic homotopy equivalence [7, 15], hence a K(1)-homology isomorphism. In other words, taking Ω^{2p} on W (ℓ) does not make any difference in K-theory. This suggests that computing K(n)_{∗}(Ω^{k}W (ℓ)) would be much easier than K(n)_{∗}(Ω^{k+3}S^{2ℓ+1}) for n ≥ 2, too.

One of the purposes of this paper is to compute K(n)_{∗}(ΩW (ℓ)) and to apply the result to determine K(n)_{∗}(Ω^{4}S^{2ℓ+1}).

There have been several attempts to determine K(n)_{∗}(Ω^{k}S^{2ℓ+1}). The case k = 1 is known as the Bott-Samelson theorem: The James splitting [4]

ΣΩS^{2ℓ+1} ≃ Σ ⋁_{j=1}^{∞} S^{2jℓ}

is multiplicative and hence h_{∗}(ΩS^{2ℓ+1}) is a polynomial algebra on a single generator for any multiplicative homology theory h_{∗}(−). We do have an analogous stable splitting for k ≥ 2, i.e. the Snaith splitting (Theorem 2.9) [11], but the summands are much more complicated and computing the Morava K-theory of each summand is not straightforward.

The Atiyah-Hirzebruch spectral sequence works fine for k = 2 [18]. The only nontrivial differential is d^{2p^{n}−1} which is essentially Milnor Q_{n} operation and the action of Q_{n} can be determined by using the Nishida relation for homology operations [9, 14]. But for k ≥ 3, we can no longer expect this simlicity. In fact, Langsetmo and Thompson [6] proved that the Atiyah-Hirzebruch spectral sequence for K(1)_{∗}(Ω^{3}S^{2ℓ+1}) never collapses at any finite stage.

The cobar-type Eilenberg-Moore spectral sequence (cobar EMSS, for short) constructed in [12] is more effective. It converges to h_{∗}(Ω^{k}Σ^{k}X) for any homology theory h_{∗}(−) and a space X with a good base point. The E^{2}-term is

E^{2} ≅ Cotor^{h_{∗}(Ω^{k−1}Σ^{k}X)}(h_{∗}, h_{∗})

when h_{∗}(−) is multiplicative and h_{∗}(Ω^{k−1}Σ^{k}X) is h_{∗}-flat. This spectral sequence can be used to compute K(n)_{∗}(Ω^{k}S^{2ℓ+1}) for k ≤ 3. In that range of k, all nontrivial differentials can be determined by the first nontrivial Atiyah-Hirzebruch differentials, i.e. Milnor Q_{n}. But difficulties arise when k > 3. Langsetmo’s work on K(1)_{∗}(Ω^{k}S^{2ℓ+1}) [5] implies that, when h_{∗}(−) = K(1)_{∗}(−), d^{2p^{2k−2}−1} ≠ 0 in the cobar spectral sequence converging to K(1)_{∗}(Ω^{k}S^{2ℓ+1}). On the other hand, K(1)_{∗}(Ω^{k}W (ℓ)) can be easily computed by using the cobar EMSS for the path-loop fibration

Ω^{k}W (ℓ) ⟶ P Ω^{k−1}W (ℓ) ⟶ Ω^{k−1}W (ℓ),

if the spectral sequence were to converge.

In this paper, we construct a strong convergent cobar EMSS for the pathloop fibration

W (n, k; ΣX) ≃ ΩW (n−1, k; Σ^{2}X) ⟶ P W (n−1, k; Σ^{2}X) ⟶ W (n−1, k; Σ^{2}X)

(1)

where W (n, k; ΣX) is the fiber of the k-fold iterated Freudenthal suspension map

E^{k} : Ω^{n−1}Σ^{n}X ⟶ Ω^{n+k−1}Σ^{n+k}X.

Theorem 1.1. For a path-connected pointed space X with (X, ∗) a strong NDR pair, there exists a spectral sequence with the following properties:

1. It strongly converges to h_{∗}(W (n, k; ΣX)) for any homology theory h_{∗}(−).

2. It is natural with respect to X and h_{∗}(−).

3. It is isomorphic to the cobar EMSS for the fibration (1) from the E^{2}-term on when h_{∗}(−) is multiplicative and h_{∗}(W (n − 1, k; Σ^{2}X)) is h_{∗}-flat.

4. The spectral sequence splits into a direct sum of spectral sequences corresponding to the stable summands of W (n, k; ΣX) under the stable splitting proved in [17].

Our construction is based on the same idea used in [12] to construct a strong convergent cobar EMSS for

Ω^{n}Σ^{n}X ⟶ P Ω^{n−1}Σ^{n}X ⟶ Ω^{n−1}Σ^{n}X.

The spectral sequence in [12] is defined by using May’s quasifibration [8]

C_{n}(X) ⟶ E_{n}(CX, X) ⟶ C_{n−1}(ΣX)

which is weakly equivalent to the above path-loop fibration together with the Snaith splittings of C_{n}(X) and E_{n}(CX, X). The spectral sequence is induced by a filtration, called the gravity filtration, on each summand of the Snaith splitting.

In order to find an analogous stable filtration on W (n, k; ΣX), we use the little cube model ξ_{n,k}(ΣX) for W (n, k; ΣX) found by Wong [17]. We extend his construction to a quasifibration

ξ_{n,k}(ΣX) ⟶ ζ_{n−1,k}(CΣX, ΣX) ⟶ ξ_{n−1,k}(Σ^{2}X)

which is weakly equivalent to the path-loop fibration (1) and then define a filtration on ζ_{n−1,k}(CΣX, ΣX) (in the stable homotopy category) to obtain a spectral sequence.

Wong proves a weak equivalence

ξ_{n,k}(ΣX) ≃_{w} W (n, k, ΣX)

by showing that there exists a quasifibration

C_{n+k}X ⟶ ξ_{n,k}(ΣX) ⟶ C_{n−1}(ΣX).  (2)

weakly equivalent to

Ω^{n+k}Σ^{n+k}X ⟶ W (n, k; ΣX) ⟶ Ω^{n−1}Σ^{n}X.  (3)

This fact suggests that the cobar spectral sequence for (3) might also converge. In fact, this is the case, although we do not use Wong’s quasifibration in the construction of the spectral sequence.

Theorem 1.2. For a path-connected pointed space X with (X, ∗) a strong NDR pair, there exists a spectral sequence with the following properties:

1. It strongly converges to h_{∗}(Ω^{n+k}Σ^{n+k}X) for any homology theory h_{∗}(−).

2. It is natural with respect to X and h_{∗}(−).

3. It is isomorphic to the cobar EMSS for the path-loop fibration

Ω^{n+k}Σ^{n+k}X ⟶ W (n, k; ΣX) ⟶ Ω^{n−1}Σ^{n}X

from the E^{2}-term on when h_{∗}(−) is multiplicative and h_{∗}(W (n, k; ΣX)) and h_{∗}(Ω^{n−1}Σ^{n}X) are h_{∗}-flat.

4. The spectral sequence splits into a direct sum of spectral sequences corresponding to the stable summands of W (n, k; ΣX) and Ω^{n+k}Σ^{n+k}X.

The paper is organazed as follows: In §2, we recall various important facts used in later sections, including iterated loop spaces and their little cube models, cobar EMSS and v_{n}-Bockstein operations. In §3, a little cube model for the path-loop fibration on the fiber of iterated Freudenthal suspension map is introduced. It is used in §4 to define the spectral sequence in Theorem 1.1. Theorem 1.2 is proved in §5. As applications of our spectral sequences, we compute K(n)_{∗}(ΩW (ℓ)) and K(n)_{∗}(Ω^{4}S^{2ℓ+1}) for an odd prime p in §6. In §7, several remarks are made.

Acknowledgements: This work could not be completed without stimulating discussions with Steve Wilson and Lisa Langsetmo, who not only tried to convince the author that K(n)_{∗}(Ω^{k}S^{2ℓ+1}) is computable and that the cobar spectral sequence for W (ℓ) converges but also provided their grants to make his visit to the Johns Hopkins University possible.

The main part of this article was written during a one-month visit to the Johns Hopkins University in December, 2000.

2 Recollections

Let us recall some important definitions and properites related to little cubes and cobar-type Eilenbergo-Moore spectral sequece.

2.1 Little Cubes

Definition 2.1. A little n-cube is an embedding of I^{n} into I^{n}

c : I^{n} ↪ I^{n}

of the form c = ℓ_{1} × ⋯ × ℓ_{n} where each ℓ_{i} is an orientation preserving linear embedding

ℓ_{i} : I ⟶ I.

The space of little n-cubes, the space of j little n-cubes and the space of nonoverlapping j little n-cubes are denoted by C_{n}(1), C_{n}^{∞}(j) and C_{n}(j), respectively.

C_{n}(1) = {c : I^{n} ⟶ I^{n}|little n-cube}
C_{n}^{∞}(j) = C_{n}(1)^{j}
C_{n}(j) = {(c_{1}, ⋯, c_{j}) ∈ C_{n}^{∞}(j)|c_{i_{1}}(IntI^{n}) ∩ c_{i_{2}}(IntI^{n}) = ∅ for i_{1} ≠ i_{2}}

For a pointed space X, define C_{n}^{∞}(X) by

C_{n}^{∞}(X) = (∐ C_{n}^{∞}(j) ×_{Σ_{j}} X^{j}) / ∼

where ∼ is the standard basepoint relation. C_{n}(X) is defined similarly by using C_{n}(j) instead of C_{n}^{∞}(j).

Note that C_{n}(X) is a subspace of C_{n}^{∞}(X).

Definition 2.2. For each 1 ≤ i ≤ n, define

p_{i} : C_{n}^{∞}(1) ⟶ C_{n−1}^{∞}(1)

by removing the i-th coordinate. This map induces a Σ_{j}-equivariant map

p_{i,j} = (p_{i})^{j} : C_{n}^{∞}(j) ⟶ C_{n−1}^{∞}(j)

and thus

π_{i} : C_{n}^{∞}(X) ⟶ C_{n−1}^{∞}(X).

For a based subspace A of X, define

π_{i,A} : C_{n}(X) ⟶ C_{n−1}^{∞}(X/A)

by the composition

C_{n}(X) ↪ C_{n}^{∞}(X) ⟶ C_{n}^{∞}(X/A) ⟶^{π_{i}} C_{n−1}^{∞}(X/A).

For each 1 ≤ i ≤ n, define a subspace E_{n,i}(X, A) of C_{n}(X) by the following pull-back diagram:

[figure: E_{n,i}(X, A); C_{n}(X); C_{n−1}(X/A); C_{n−1}^{∞}(X/A); π_{i,A}]

The restriction of π_{i,A} to E_{n,i}(X, A) is also denoted by π_{i,A}. We usually denote E_{n,1}(X, A) by E_{n}(X, A) and π_{1,A} by π_{A}.

Remark 2.3. It is useful to write an element in E_{n,i}(X, A) as [c, d; x, a] where

c ∈ C_{n}(j_{1})
d ∈ C_{n}(j_{2})
x ∈ (X − A)^{j_{1}}
a ∈ A^{j_{2}}

and the cubes in c can be extended to the right-hand side in the i-th coordinate of the cubes without intersecting other cubes.

In the rest of this paper, we always use a representative of the above form for elements in E_{n,i}(X, A).

May [8] proved the following:

Theorem 2.4 (May). If (X, A) is a strong NDR pair and A is connected, then

π_{A} : E_{n}(X, A) ⟶ C_{n−1}(X/A)

is a quasifibration with fiber C_{n}(A).

This quasifibration is weakly equivalent to the path-loop fibration on Ω^{n−1}Σ^{n}Y when (X, A) = (CY, Y).

Theorem 2.5 (May). There exist natural maps

α_{n} : C_{n}(Y) ⟶ Ω^{n}Σ^{n}Y
α̃_{n} : E_{n}(CY, Y) ⟶ P Ω^{n−1}Σ^{n}Y

which make the following diagram commutative

[figure: C_{n}(Y); E_{n}(CY, Y); C_{n−1}(ΣY); Ω^{n}Σ^{n}Y; P Ω^{n−1}Σ^{n}Y; Ω^{n−1}Σ^{n}Y.; α_{n}; α̃_{n}; α_{n−1}]

If Y is connected and (Y, ∗) is a strong NDR pair, these maps are weak homotopy equivalences.

The above weak equivalence is multiplicative in the following sense.

Definition 2.6. For 1 ≤ i ≤ n and a pointed space X, define

μ_{i} : C_{n}(X) × C_{n}(X) ⟶ C_{n}(X)

by concatenating little cubes in the i-th coordinate. We usually denote μ_{1} by μ.

Lemma 2.7. μ_{i} coincides with the H-space multiplication in the i-th loop on Ω^{n}Σ^{n}X under the weak equivalence α_{n}.

One of the most important properties of C_{n}(X) besides the above May’s theorem is the Snaith splitting [11].

Definition 2.8. For a pointed space X, define

F_{k}C_{n}(X) = (∐_{j=1}^{k} C_{n}(j) ×_{Σ_{j}} X^{j}) / ∼

Theorem 2.9 (Snaith). For a pointed space X with (X, ∗) a strong NDR pair, we have the following natural stable homotopy equivalence

C_{n}(X) ≃_{S} ⋁_{j=1}^{∞} F_{j}C_{n}(X)/F_{j−1}C_{n}(X)

= ⋁_{j=1}^{∞} C_{n}(j) ∧_{Σ_{j}} X^{∧j}.

The above splitting is also known to be multiplicative.

Proposition 2.10. For a pointed space X with (X, ∗) a strong NDR pair, we have the following commutative diagram

[figure: C_{n}(X) ∧ C_{n}(X); C_{n}(X); ( ⋁_{j=1}^{∞} C_{n}(j) ∧_{Σ_{j}} X^{∧j}) ∧ ( ⋁_{j=1}^{∞} C_{n}(j) ∧_{Σ_{j}} X^{∧j}); ⋁_{j=1}^{∞} C_{n}(j) ∧_{Σ_{j}} X^{∧j}; μ_{i}; ∨μ_{i}]

The construction of the spectral sequence in Theorem 1.1 is based on the following Wong’s definition.

Definition 2.11. For a pointed space X, define ξ_{n,k}(ΣX) by the following pull-back diagram

[figure: ξ_{n,k}(ΣX); E_{n+k}(CX, X); C_{n−1}(ΣX); C_{n+k−1}(ΣX); π_{n,k}; π_{X}; σ^{k}]  (4)

where σ^{k} is induced by the map

C_{n−1}(1) ⟶ C_{n+k−1}(1)

whch maps c to c × 1_{I^{k}}.

Remark 2.12. Note that the upper horizontal map in the diagram (4) is injective and ξ_{n,k}(ΣX) can be considered to be a subspace of E_{n+k}(CX, X). With this identification, an element in ξ_{n,k}(Σ, X) can be written as [c × 1_{I^{k}}, d; x, a] where

c ∈ C_{n}(j_{1})
d ∈ C_{n}(j_{2})
x ∈ (CX − X)^{j_{1}}
a ∈ X^{j_{2}},

under the notation in Remark 2.3.

ξ_{n,k}(ΣX) was defined to serve as a little cube model for W (n, k; ΣX).

Theorem 2.13 (Wong). For a path-connected pointed space X with (X, ∗) a strong NDR pair,

π_{n,k} : ξ_{n,k}(ΣX) ⟶ C_{n−1}(ΣX)

in diagram (4) is a quasifibration with fiber C_{n+k}(X).

The restriction of α̃_{n+k} gives us a natural weak homotopy equivalence

α̃_{n,k} : ξ_{n,k}(ΣX) ⟶ W (n, k; ΣX)

which makes the following diagram commutative

[figure: C_{n+k}(X); ξ_{n,k}(ΣX); C_{n−1}(ΣX); Ω^{n+k}Σ^{n+k}X; W (n, k; ΣX); Ω^{n−1}Σ^{n}X; α_{n+k}; α̃_{n,k}; α_{n−1}]

Wong [17] proves the existence of a stable splitting of ξ_{n,k}(ΣX) generalizing the Snaith splitting of C_{n}(X). More generally the following is a useful criterion for the existence of stable splitting.

Lemma 2.14 (Wong). Let X be a pointed space with (X, ∗) a strong NDR pair and ξ ⊂ C_{n}(X) be a subspace with the following properties:

1. Let F_{j}ξ = F_{j}C_{n}(X) ∩ ξ. Then (F_{j}ξ, F_{j−1}ξ) is a strong NDR pair for all j.

2. For any [c, x] ∈ F_{j}ξ, if [c̃, x̃] is an element of ξ obtained from [c, x] by removing little cubes and corresponding points in X, then [c̃, x̃] ∈ F_{q}ξ, where q is the number of little cubes in c̃. then the Snaith splittiing map on C_{n}(X) restricts to ξ to give a stable splitting map

ξ ≃_{S} ⋁_{j=1}^{∞} F_{j}ξ/F_{j−1}ξ.

Corollary 2.15 (Wong). Under the same condition as in the above lemma, we have the following stable homotopy equivalence

ξ_{n,k}(ΣX) ≃_{S} ⋁_{j=1}^{∞} F_{j}ξ_{n,k}(ΣX)/F_{j−1}ξ_{n,k}(ΣX)

which is compatible with the Snaith splitting map on C_{n+k}(X) under the map

C_{n+k}(X) ⟶ ξ_{n,k}(ΣX).

2.2 Eilenberg-Moore Spectral Sequence

In order to prove that our spectral sequences in Theorem 1.1 and Theorem 1.2 are isomorphic to the corresponding classical Eilenberg-Moore spectral sequences, we need to make use of the notion of display introduced in [10].

Definition 2.16. For a topological space B, let (Spaces /B)_{∗} be the category of pointed spaces over B. An object in (Spaces /B)_{∗} is a continuous map

f : X ⟶ B

together with a cross section

s : B ⟶ X.

For a homology theory h_{∗}(−) on the category of topological spaces, define

h_{∗}^{B}(f, s) = h̃_{∗}(X/s(B)).

h_{∗}^{B}(−) can be extended to a functor on (Spaces /B)_{∗}.

Definition 2.17. Let (f, s) be an object in (Spaces /B)_{∗}. An h_{∗}^{B}-display of (f, s) is a sequence of cofibrations

(f, s)  ⟶^{α_{0}} (h_{0}, u_{0}) ⟶^{β_{−1}}  (f_{−1}, s_{−1})
(f_{−1}, s_{−1}) ⟶^{α_{−1}} (h_{−1}, u_{−1}) ⟶^{β_{−2}}  (f_{−2}, s_{−2})
⋮
(f_{−i}, s_{−i}) ⟶^{α_{−i}} (h_{−i}, u_{−i}) ⟶^{β_{−i−1}} (f_{−i−1}, s_{−i−1})
⋮

satisfying the following two conditions.

1. h_{∗}^{B}(h_{−i}, u_{−i}) is a flat h_{∗}-module and an injective h̃_{∗}(B_{+})-comodule for each i.

2. α_{−i}_{∗} : h_{∗}^{B}(f_{−i}, s_{−i}) ⟶ h_{∗}^{B}(h_{−i}, u_{−i}) is a monomorphism.

We also need the following notation.

Definition 2.18. Let Spaces /B be the category of spaces over B and define a functor

G : Spaces /B ⟶ (Spaces /B)_{∗}

by

G(X ⟶^{f} B) = (X ∐ B \underrightarrow{f ∐ 1_{B}} B) .

The following fact is proved in [13].

Proposition 2.19. Suppose we have a pull-back diagram of pointed spaces

[figure: F; E; {∗}; B.; p; ι_{0}]

together with a decreasing filtration on E.

⋯ ⊂ F_{−q−1}E ⊂ F_{−q}E ⊂ ⋯ ⊂ F_{−1}E ⊂ F_{0}E = E.

Let {F_{−q}F } be the induced filtration on F and suppose each consecutive inclusions

F_{−q−1}E ⟶^{i_{−q−1}} F_{−q}E
F_{−q−1}F ⟶^{i_{−q−1}} F_{−q}F

are pointed cofibrations.

Given the above data, let p_{−q} : F_{−q}E ⟶ B be the restriction of p on F_{−q}E and h_{∗}(−) be a homology theory. If

i_{−q−1} : F_{−q−1}E ⟶ F_{−q}E

induces an injective map in h_{∗}(−) and

h̃_{∗}(F_{−q}E/F_{−q−1}E) ≅ h̃_{∗}(F_{−q}F/F_{−q−1}F) ⊗_{h_{∗}} h̃_{∗}(B_{+})

as h̃_{∗}(B_{+})-comodules, then

G(p)  ⟶  C(G(i_{−1}))  ⟶  ΣG(p_{−1})
ΣG(p_{−1}) ⟶ ΣC(G(i_{−2})) ⟶  Σ^{2}G(p_{−2})

⋮

Σ^{q}G(p_{−q}) ⟶ Σ^{q}C(G(i_{−q−1})) ⟶ Σ^{q+1}G(p_{−q−1})

⋮ is an injective h_{∗}^{B}-display of G(p) with respect to G(ι_{0}). And the stable filtration on F induced by this display coincides with the one induced directly from the filtration on E.

In particuler, under the above conditions the spectral seqeunce defined by the filtration on F

E_{−q,∗}^{1} = h̃_{∗}(F_{−q}F/F_{−q−1}F) ⟹ h̃_{∗}(F)

is isomorphic to the classical Eilenberg-Moore spectral sequence for the fibration

F ⟶ E ⟶ B

from the E^{2}-term on.

In order to perform computations in §6, we need the following.

Lemma 2.20. Let E(x) be the exterior algebra over a field k generated by x, then

Cotor^{E(x)}(k, k) ≅ P (τ(x))

where τ(x) is represented in the cobar construction by [x] and called the transgression of x.

Lemma 2.21. Let T Γ_{a}(x) be the truncated divided power algebra of height p^{a} over a field k of characteristic odd prime p, i.e. the coalgebra dual to the truncated polynomial algebra of height p^{a} T P_{a}(x^{∗}). Then

Cotor^{T Γ_{a}(x)}(k, k) ≅ E(τ(x)) ⊗ P (βQ_{1}^{a}τ(x))

as algebras where τ(x) is as above and βQ_{1}^{a}τ(x) is the element represented in the cobar construction by [x^{p^{a−1}}|x^{p^{a−1}(p−1)}].

The above notation βQ_{1} is based on the well-known relation between homology operation in singular homology and the cobar spectral sequence. We also use the following notations.

Definition 2.22. In a commutative algebra C over a field k of characteristic p, x^{p} is denoted by Q_{0}(x). Furthermore, if C is a bicommutative Hopf algebra, the transgression of Q_{0}(x) in Cotor^{C}(k, k) is denoted by Q_{1}(τ(x)).

Inductively, the transgression of Q_{i}(x) is denoted by Q_{i+1}(τ(x)).

Remark 2.23. The cobar spectral sequence for

Ω^{ℓ}Σ^{ℓ}X ⟶ P Ω^{ℓ−1}Σ^{ℓ}X ⟶ Ω^{ℓ−1}Σ^{ℓ}X

with coefficients in ordinary mod p homology collapses at E^{2}. And the above notations coincide with the homology operations in H_{∗}(Ω^{ℓ}Σ^{ℓ}X).

2.3 The v_{n}-Bockstein operation

In order to compute the differentials in our cobar spectral sequences, we compare them with the Atiyah-Hirzebruch spectral sequences for the n-th Morava Ktheory.

It is a well-known fact that the first nontrivial differential in the Atiyah-Hirzebruch spectral sequence for the n-th Morava K-theory

E^{2} ≅ H_{∗}(Y; ℤ/pℤ) ⊗ K(n)_{∗} ⟹ K(n)_{∗}(Y)

is given by Milnor’s Q_{n}-operation. In order to avoid confusions, we denote Milnor’s Q_{n}-operation by β_{v_{n}}^{1}. Thus the first nontrivial differential in the above spectral sequence is given by

d^{2p^{n}−1}(y ⊗ 1) = β_{v_{n}}^{1}y ⊗ v_{n}.

The following formula is proved in [14].

Proposition 2.24. For a path-connected space X and n > 0, the following formula holds in H_{∗}(Ω^{ℓ}Σ^{ℓ}X): For x ∈ H̃_{∗}(X), when 0 ≤ i ≤ ℓ − 2 or X = S^{2m+1}, we have

β_{v_{n}}^{1}Q_{i}(x) = Q_{i−1}(β_{v_{n−1}}^{1}(x)) − Q_{i−2p^{n−1}−1}(β_{v_{n−2}}^{1}(x)) + ⋯
+(−1)^{n−1}Q_{i−2p^{n−1}−2p^{n−2}−···−2p−1}(β_{v_{0}}^{1}(x))
+(−1)^{n}βQ_{i−2p^{n−1}−···−2p−2}(x)

where d_{n,k} = \frac{2p^{n}−2p^{k}}{p−1} + 1 for k > 0 and d_{n} = \frac{2p^{n}−2}{p−1}. By applying β, we obtain

β_{v_{n}}^{1}βQ_{i}(x) = ∑_{k=1}^{n}(−1)^{n−k+1}βQ_{i−d_{n,k}}(β_{v_{k−1}}^{1}(x)).

3 The Little Cube Models

The spectral sequence in [12] was constructed by using the quasifibration

C_{n}(X) ⟶ E_{n}(CX, X) ⟶ C_{n−1}(ΣX)

together with stable splittings of C_{n}(X) and E_{n}(CX, X) which are compatible with the inclusion maps

[figure: C_{n}(X); E_{n}(CX, X); ⋁_{j=1}^{∞} F_{j}C_{n}(X)/F_{j−1}C_{n}(X); ⋁_{j=1}^{∞} F_{j}E_{n}(CX, X)/F_{j−1}E_{n}(CX, X); ≃_{S}; ≃_{S}]

Let us denote F_{j}E_{n}(CX, X)/F_{j−1}E_{n}(CX, X) by Ē_{n}(j; CX, X). Theorem 3.1 ([12]). There exist decreasing filtrations on C_{n}(j) and Ē_{n}(j; CX, X)

∅ = F_{−j−1}C_{n}(j) ⊂ F_{−j}C_{n}(j) ⊂ ⋯ ⊂ F_{−1}C_{n}(j) = F_{0}C_{n}(j) = C_{n}(j)
∅ = F_{−j−1}Ē_{n,j} ⊂ F_{−j}Ē_{n,j} ⊂ ⋯ ⊂ F_{−1}Ē_{n,j} ⊂ F_{0}Ē_{n,j} = Ē_{n}(j; CX, X)

with the following properties.

1. The filtration on C_{n}(j) is Σ_{j}-equivariant, hence induces a filtration on C_{n}(j)_{+} ∧_{Σ_{j}} X^{∧j}.

2. When X is path-connected, the inclusions

F_{−q−1}C_{n}(j)_{+} ∧_{Σ_{j}} X^{∧j} ⊂ F_{−q}C_{n}(j)_{+} ∧_{Σ_{j}} X^{∧j}
F_{−q−1}Ē_{n,j} ⊂ F_{−q}Ē_{n,j}

are poinited cofibrations for each q.

3. The inclusion

C_{n}(j)_{+} ∧_{Σ_{j}} X^{∧j} ⊂ Ē_{n}(j; CX, X)

is filtration preserving.

4. Define

F_{−q}C_{n}(X) = ⋁_{j=1}^{∞} F_{−q}C_{n}(j)_{+} ∧_{Σ_{j}} X^{∧j}
F_{−q}E_{n}(CX, X) = ⋁_{j=1}^{∞} F_{−q}Ē_{n}(j; CX, X).

Then, for a path-connected space X, we have a stable homotopy equivalence

F_{−q}E_{n}(CX, X)/F_{−q−1}E_{n}(CX, X) ≃_{S} (F_{−q}C_{n}(X)/F_{−q−1}C_{n}(X))∧C_{n−1}(ΣX)

for each q. The spectral sequence in [12] is the spectral sequence induced by the above

stable filtration on C_{n}(X). The last property gurantees that the spectral sequence is isomorphic to the classical Eilenberg-Moore spectral sequence [13].

In order to apply this method to prove Theorem 1.1, we first need a quasifibration made out of little cubes which is weakly equivalent to the path-loop fibration

W (n + 1, k; ΣX) ⟶ P W (n, k; Σ^{2}X) ⟶ W (n, k; Σ^{2}X)

and then stable splittings of the total space and the fiber of the quasifibration.

Consider the following diagram of fibrations.

[figure: P Ω^{n+k}Σ^{n+k+1}X; W (n + 1, k; ΣX); Ω^{n+k}Σ^{n+k+1}X; Ω^{n}Σ^{n+1}X; P^{2}Ω^{n+k−1}Σ^{n+k+1}X; P W (n, k; Σ^{2}X); P Ω^{n+k−1}Σ^{n+k+1}X; P Ω^{n−1}Σ^{n+1}X; P Ω^{n+k−1}Σ^{n+k+1}X; W (n, k; Σ^{2}X); Ω^{n+k−1}Σ^{n+k+1}X; Ω^{n−1}Σ^{n+1}X]

Every vertical sequence is a fibration and each horizontal square is a pullback diagram. The corresponding diagram of spaces made of little cubes would be

[figure: E_{n+k+1,2}(CX, X); ξ_{n+1,k}(ΣX); C_{n+k}(ΣX); C_{n}(ΣX); ?_{1}; ?_{2}; E_{n+k,2}(CΣX, ΣX); E_{n,2}(CΣX, ΣX); E_{n+k,1}(CΣX, ΣX); ξ_{n,k}(Σ^{2}X); C_{n+k−1}(Σ^{2}X); C_{n−1}(Σ^{2}X)]

Once we obtain a little cube model for the fibration

P Ω^{n+k}Σ^{n+k+1}X ⟶ P^{2}Ω^{n+k−1}Σ^{n+k+1}X ⟶ P Ω^{n+k−1}Σ^{n+k+1}X,

i.e. a little cube model ?_{1} for P^{2}Ω^{n+k−1}Σ^{n+k+1}X in the above diagram, we would get a little cube model ?_{2} for P W (n − 1, k; Σ^{2}X) as the pull-back of the middle horizontal square.

We will work in more general settings to get better insights.

Definition 3.2. For a triad of spaces (X; A, B), define

EE_{n}(X; A, B) = E_{n,1}(X, A) ∩ E_{n,2}(X, B).

Theorem 3.3. Let (X, A) be an NDR pair with an NDR representation (h, u) and B be a subspace of X with h(B × I) ⊂ B. If both A and A ∩ B is pathconnected, then the followiing is a quasifibration

E_{n,2}(A, A ∩ B) ⟶ EE_{n}(X; A, B) ⟶^{π_{A,1}} E_{n−1,1}(X/A, B/A ∩ B).

As is usually the case when one wants to prove a map is a quasifibration, we need the following theorem due originally to Dold and Thom [3] and restated in the following form in [8], known as the Dold-Thom criterion.

Theorem 3.4. Let B be a filtered space

F_{0}B ⊂ F_{1}B ⊂ ⋯ ⊂ B

and

p : E ⟶ B

be a surjective map. p is a quasifibration if the following conditions hold:

1. The restrictions of p to F_{0}B and any open subset of F_{j}B − F_{j−1}B for j > 0 are quasifibrations.

2. For each j > 0, there exist an open neighborhood U of F_{j−1}B in F_{j}B and

h : U × I ⟶ U
H : p^{−1}(U) × I ⟶ p^{−1}(U)

with the following properties

(a) p ∘ H = h ∘ p.

(b) h and H are deformations of U and p^{−1}(U) into F_{j−1}B and p^{−1}(F_{j−1}B), respectively.

(c) H_{1} is a weak homotopy equivalence on each fiber. Proof of Theorem 3.3. Note that

π_{A,1}(E_{n,1}(X, A) ∩ E_{n,2}(X, B)) = E_{n−1}(X/A, B/A ∩ B)

and thus the projection

π_{A,1} : E_{n}(X, A) ⟶ C_{n−1}(X/A)

restricts to give us a surjective map

π_{A} : EE_{n}(X; A, B) ⟶ E_{n−1}(X/A, B/A ∩ B).

Furthermore

π_{A}^{−1}(∗) = E_{n,1}(X, A) ∩ E_{n,2}(X, B) ∩ C_{n}(A)
= E_{n,2}(X, B) ∩ C_{n}(A)
= E_{n,2}(A, A ∩ B).

It remains to check that the filtration on E_{n−1}(X/A, B/A ∩ B) defined by F_{j}E_{n−1}(X/A, B/A ∩ B) = F_{j}C_{n−1}(X/A) ∩ E_{n−1}(X/A, B/A ∩ B) satisfies the Dold-Thom criterion. Note that we have the following commutative diagram

[figure: E_{n,2}(A, A ∩ B); C_{n}(A); EE_{n}(X; A, B); E_{n}(X, A); E_{n−1,1}(X/A, B/A ∩ B); C_{n−1}(X/A)]

May [8] proved that the right-hand side satisfies the Dold-Thom criterion by defining various maps and homotopies. All we have to do is to check that those maps and homotopies can be restricted to our spaces. The following is the list of maps May used.

1. For U ⊂ F_{j}C_{n}(X/A) − F_{j−1}C_{n}(X/A),

(a)

\begin{matrix}π_{A}^{−1}(U) & ⟶^{q} & C_{n}(A) \\ [c, d; x, a] & ⟼ & [d, a] \end{matrix}

(b)

U × C_{n}(A) ⟶^{w}  π_{A}^{−1}(U)
([c, x], [d, a]) ⟼ μ_{1}([c, x], [d, a])

where

μ_{1} : C_{n}(X) × C_{n}(X) ⟶ C_{n}(X)

is the multiplication by concatenating cubes in the first coordinates.

(c) A homotopy

G^{−} : U × C_{n}(A) × I ⟶ U × C_{n}(A)

giving a fiberwise homotopy

(π_{n} × q) ∘ w ≃ 1_{U×C_{n}(A)}.

G^{−} is given by shrinking the first coordinates of the cubes in C_{n}(A) into [0, \frac{1}{2}].

(d) A homotopy

G^{+} : π_{A}^{−1}(U) × I ⟶ π_{A}^{−1}(U)

giving a fiberwise homotopy

w ∘ (π_{n} × q) ≃ 1_{π_{A}^{−1}(U)}.

This is also a deformation of cubes in the first coordinates.

2. Let (ũ_{j}, h̃_{j}) be the strong NDR representation for (F_{j}C_{n}(X/A), F_{j−1}C_{n}(X/A)) defined by the strong NDR representation for (X, A) in [8]. And let

V_{j} = {[c, x] ∈ F_{j}C_{n}(X/A)|ũ_{j}([c, x]) < 1}.

(a) h̃_{j} gives us a deformation

h̃_{j} : V_{j} × I ⟶ V_{j}.

(b) A deformation

ℓ̃_{j} : π_{A}^{−1}(V_{j}) × I ⟶ π_{A}^{−1}(V_{j})

is defined by

(ℓ̃_{j})_{t}|_{π_{A}^{−1}(F_{j−1}C_{n}(X/A))} = id

and

(ℓ̃_{j})_{t}|_{π_{A}^{−1}(V_{j})−π_{A}^{−1}(F_{j−1}C_{n}(X/A))}([c, d; x, a]) = [c, d; h_{j}(x, t), a]

3. Finally May showed that

(ℓ̃_{j})_{1} : π_{A}^{−1}([c, x]) ⟶ π_{A}^{−1}((h̃_{j})_{1}([c, x]))

is a homotopy equivalence by using a homotopy

H : (([c, x]) × C_{n}(A)) × I ⟶ π_{A}^{−1}((h̃_{j})_{1}([c, x]))

giving

(ℓ̃_{j})_{1} ∘ w ≃ w ∘ ((h̃_{j})_{1} × 1).

H is defined by using a deformation of cubes in the first coordinates and paths in A connecting points belonging to A in h_{j}(x) to the base point.

Note that in our definition of EE_{n}(X; A, B) the condition on the first coordinate of cubes is exactly same as that of E_{n}(X, A). Thus the maps from 1(a) to 1(d) can be safely restricted to our spaces, since they only involves in the first coordinates of cubes.

For 2(a) and 2(b), the assumption on the NDR representation (h, u) for (X, A), i.e. h(B × I) ⊂ B gurantees that these homotopies can be restricted to our spaces.

In the definition of the homotopy in 3, we can choose a path in A ∩ B for points in A∩B because of our assumption on the path-connectivity of A∩B.

In order to obtain a model for a path-loop fibration, we need a quasifibration with contractible total space.

Proposition 3.5. Let X be a pointed space with (X, ∗) a strong NDR pair, then EE_{n}(C^{2}X; C_{1}X, C_{2}X) is contractible, where

C^{2}X = X ∧ I ∧ I
C_{1}X = X ∧ I ∧ S^{0}
C_{2}X = X ∧ S^{0} ∧ I.

Proof. May [8] defines a contraction of E_{n}(CX, X) by using the contraction of CX together with a strong NDR representation of (X, ∗).

All we have to do here is to make sure that there exists a contraction of C^{2}X which keeps C_{2}X in itself. But a contraction on C_{1}X naturally induces such a contraction on C^{2}X = C_{1}X ∧ I.

Thus we have a quasifibration

E_{n,2}(CX, X) ⟶ EE_{n}(C^{2}X; C_{1}X, C_{2}X) ⟶^{π_{CX,1}} E_{n−1,1}(CΣX, ΣX)

consisting of contractible spaces. This is our model for the path-loop fibration on P Ω^{n−1}Σ^{n+1}X.

Definition 3.6. For a pointed space X, define ζ_{n,k}(CΣX, ΣX) by the following pull-back diagram

[figure: ζ_{n,k}(CΣX, ΣX); EE_{n+k+1}(C^{2}X; CX, CX); E_{n}(CΣX, ΣX); E_{n+k}(CΣX, ΣX); E^{k}]

Proposition 3.7. ζ_{n,k}(CΣX, ΣX) is contractible for X with strongly nondegenerate basepoint.

Proof. The contraction in Proposition 3.5 restricts to give a contraction of ζ_{n,k}(CΣX, ΣX).

Theorem 3.8. For a path-connected pointed space X with (X, ∗) strong NDR pair, the following is a quasifibration

ξ_{n+1,k}(ΣX) ⟶ ζ_{n,k}(CΣX, ΣX) ⟶ ξ_{n,k}(Σ^{2}X).

Furthermore Wong’s equivalence

α̃ : ξ_{n,k}(ΣX) ⟶ W (n, k; ΣX)

induces a weak equivalence of quasifibrations

[figure: ξ_{n+1,k}(ΣX); ζ_{n,k}(CΣX, ΣX); ξ_{n,k}(Σ^{2}X); W (n + 1, k)(ΣX); P W (n, k; Σ^{2}X); W (n, k; Σ^{2}X).; ≃_{w}; ≃_{w}; ≃_{w}]

Proof. We have the following commutative diagram.

[figure: ξ_{n+1,k}(ΣX); E_{n+k+1,2}(CX, X); ζ_{n,k}(ΣX); EE_{n+k+1}(C^{2}X, C_{1}X, C_{2}X); ξ_{n,k}(Σ^{2}X); E_{n+k,1}(CΣX, ΣX).]

The maps and homotopies used in the proof that the right hand side is a quasifibration can be used to prove that the left hand side satisfies the Dold-Thom criterion. The details are omitted.

4 The Gravity Filtration on ξ_{n,k}(ΣX)

The purpose of this section is to define a stable filtration on ξ_{n,k}(ΣX) and ζ_{n,k}(CΣX, ΣX) which induce a cobar spectral sequences for the following pullback diagram

[figure: W (n + 1, k; ΣX); P W (n, k; Σ^{2}X); ∗; W (n, k; Σ^{2}X)]

As a corollay to Lemma 2.14, we obtain the following.

Corollary 4.1. For a pointed space X with strongly nondegenrate basepoint, we have the following stable homotopy equivalence

ζ_{n,k}(CΣX, ΣX) ≃_{S} ⋁_{j=1}^{∞} F_{j}ζ_{n,k}(CΣX, ΣX)/F_{j−1}ζ_{n,k}(CΣX, ΣX)

which is compatible with the Snaith splitting map on ξ_{n,k}(ΣX) under the inclusion

ξ_{n,k}(ΣX) ⟶ ζ_{n,k}(CΣX, ΣX).

Note that we have the following sequence of inclusions

ζ_{n,k}(CΣX, ΣX) ⊂ EE_{n+k+1}(C^{2}X, CX, CX) ⊂ E_{n+k+1}(C^{2}X, CX) which preserve the Snaith filtrations. We define a stable filtration on ζ_{n,k}(CΣX, ΣX) by restricting the gravity filtration on E_{n+k+1}(C^{2}X, CX). To be more precise, let us use the following notations for simplicity

ξ̃_{n,k}(j; ΣX) = F_{j}ξ_{n,k}(ΣX)/F_{j−1}ξ_{n,k}(ΣX)
ζ̃_{n,k}(j; CΣX, ΣX) = F_{j}ζ_{n,k}(CΣX, ΣX)/F_{j−1}ζ_{n,k}(CΣX, ΣX).

Definition 4.2. For each j ≥ 1, define a filtration {F_{−q}ζ̃_{n,k}(j; CΣX, ΣX)}_{q≥0} on ζ̃_{n,k}(j; CΣX, ΣX) as the inverse image of the gravity filtration on Ẽ_{n+k+1}(j; C^{2}X, CX) under the above inclusion.

Define

F_{−q}ζ_{n,k}(CΣX, ΣX) = ⋁_{j=1}^{∞} F_{−q}ζ̃_{n,k}(j; CΣX, ΣX)

By Corollary 4.1, the inclusion

ξ_{n,k}(ΣX) ↪ ζ_{n,k}(CΣX, ΣX)

is compatible with the stable splitting maps.

Definition 4.3. For each j ≥ 1, let {F_{−q}ξ̃_{n,k}(j; ΣX)}_{q≥0} be the filtration on ξ̃_{n,k}(j; ΣX) induced by the above inclusion.

Define

F_{−q}ξ_{n,k}(ΣX) = ⋁_{j=1}^{∞} F_{−q}ξ̃_{n,k}(j; ΣX).

The maps and homotopies in the proof of Theorem 3.1 can be restricted without a modification to give us the following. The proof is omitted.

Theorem 4.4. The above filtrations

{F_{−q}ζ_{n,k}(CΣX, ΣX)}_{q≥0} and {F_{−q}ξ_{n,k}(ΣX)}_{q≥0}

on

⋁_{j=1}^{∞} ζ̃_{n,k}(j; CΣX, ΣX) and ⋁_{j=1}^{∞} ξ̃_{n,k}(j; ΣX)

satisfy the following properties:

1. The inclusions

F_{−q−1}ζ̃_{n,k}(j; CΣX, ΣX) ⊂ F_{−q}ζ̃_{n,k}(j; CΣX, ΣX)
F_{−q−1}ξ̃_{n,k}(j; ΣX) ⊂ F_{−q}ξ̃_{n,k}(j; ΣX)

are poinited cofibrations for each q.

2. The inclusion

ξ̃_{n,k}(j; ΣX) ⊂ ζ̃_{n,k}(j; CΣX, ΣX)

is filtration preserving.

3. F_{−q}ζ_{n,k}(CΣX, ΣX)/F_{−q−1}ζ_{n,k}(CΣX, ΣX) is stably homotopy equivalent

to

(F_{−q}ξ_{n,k}(ΣX)/F_{−q−1}ξ_{n,k}(ΣX)) ∧ ξ_{n−1,k}(Σ^{2}X)

for each q.

Now Theorem 1.1 follows from the above Theorem together with Proposition 2.19.

5 Filtering Ω^{n+k}Σ^{n+k}X

In this section, we define a stable filtration on Ω^{n+k}Σ^{n+k}X which induces a cobar spectral sequence for the fibration

Ω^{n+k}Σ^{n+k}X ⟶ W (n, k; ΣX) ⟶ Ω^{n−1}Σ^{n}X.  (5)

In other words, we want to find a stable filtration on Ω^{n+k}Σ^{n+k}X whose associated spectral sequence for homology theory h_{∗}(−) is of the following form

E^{2} ≅ Cotor^{h_{∗}(Ω^{n−1}Σ^{n}X)}(h_{∗}, h_{∗}(W (n, k; ΣX))),

if h_{∗}(−) is multiplicative and h_{∗}(Ω^{n−1}Σ^{n}X) is h_{∗}-flat.

One of the most natural idea would be to use the quasifibration model for (5) (Theorem 2.13)

C_{n+k}(X) ⟶ ξ_{n,k}(ΣX) ⟶ C_{n−1}(ΣX).

Unfortunately, however, C_{n+k}(X) is too small for our purpose. What we need is a space, say C_{n,k}(X) with a stable filtration {F_{−q}C_{n,k}(X)} and the following stable homotopy equivalence,

F_{−q}C_{n,k}(X)/F_{−q−1}C_{n,k}(X) ≃_{S} F_{−q}C_{n}(X)/F_{−q−1}C_{n}(X) ∧ ξ_{n,k}(ΣX).

Furthermore, we want C_{n,k}(X) to be a subspace of C_{n+k}(CX) to get a stable splitting by applying Lemma 2.14.

Definition 5.1. Define C_{n,k}(X) to be the subspace of C_{n+k}(CX) consisting of the elements of the following form:

[c × 1_{I^{k}}, μ_{n+k}(d, e × 1_{I^{k}}); x, a, b]

where

[c × 1_{I^{k}}, d; x, a] ∈ ξ_{n,k}(ΣX)
[c, e; x, b] ∈ E_{n}(CX, X)

and μ_{n+k} is the concatenation of cubes in the (n + k)-th coordinate, i.e. the last coordinate. A typical element in C_{n,k}(X) looks like the following picture.

d_{1}  d_{2} e_{1} × 1_{I^{k}}  e_{2} × 1_{I^{k}}  c_{1} × 1_{I^{k}}

Splitting little cubes into the upper and lower halves induces the following maps

π_{+} : C_{n,k}(X) ⟶ ξ_{n−1,k}(ΣX)
π_{−} : C_{n,k}(X) ⟶ E_{n}(CX, X).

Remark 5.2. The pull-back diagram

[figure: Ω^{n+k}Σ^{n+k}X; P Ω^{n−1}Σ^{n}X; W (n, k; ΣX); Ω^{n−1}Σ^{n}X]

suggests that the space C̄_{n,k}(X) defined by the pull-back diagram

[figure: C̄_{n,k}(X); E_{n}(CX, X); ξ_{n,k}(ΣX); C_{n−1}(ΣX); ĩ_{n,k}; π_{n,k}]

could be used as a little cube model for Ω^{n+k}Σ^{n+k}X. In fact, it is easy to see that the inclusion map

C_{n+k}(X) ⟶ C̄_{n,k}(X)

is a homotopy equivalence.

The above definition of C_{n,k}(X) is a slight modification of C̄_{n,k}(X) to get a stable splitting.

Proposition 5.3. π_{+} is a quasifibration with fiber C_{n}(X).

Proof. The Dold-Thom criterion (Theorem 3.4) applies.

Corollary 5.4. For a path-connected pointed space X with (X, ∗) strong NDR, we have a weak homotopy equivalence

C_{n,k}(X) ≃ Ω^{n+k}Σ^{n+k}X.

C_{n,k}(X) is our model for Ω^{n+k}Σ^{n+k}X. It satisfies the condition for Lemma 2.14 and we have a stable splitting.

Lemma 5.5. For a path-connected pointed space X with (X, ∗) strong NDR pair, the Snaith splitting map on C_{n+k}(CX) induces a stable homotopy equivalence

C_{n,k}(X) ≃_{S} ⋁_{j=1}^{∞} F_{j}C_{n,k}(X)/F_{j−1}C_{n,k}(X).

Definition 5.6. F_{j}C_{n,k}(X)/F_{j−1}C_{n,k}(X) is denoted by C̃_{n,k}(j; X).

In order to define a stable filtration on C_{n,k}(X), we make use of the map

π_{−} : C_{n,k}(X) ⟶ E_{n}(CX, X)

or the induced map

π_{−} : C̃_{n,k}(j; X) ⟶ ⋁_{1≤ℓ≤j} Ẽ_{n}(ℓ; CX, X).

Definition 5.7. Define a filtration {F_{−q}C̃_{n,k}(j; X)} on C̃_{n,k}(j; X) by

(c, x) ∈ F_{−q}C̃_{n,k}(j; X) ⟺ π_{−}(c, x) ∈ ⋁_{1≤ℓ≤j} F_{−q}Ẽ_{n}(ℓ; CX, X)

where {F_{−q}Ẽ_{n}(j; CX, X)} is the gravity filtration on Ẽ_{n}(j; CX, X) defined in [12].

And define a stable filtration on C_{n,k}(X) by

F_{−q}C_{n,k}(X) = ⋁_{j=1}^{∞} F_{−q}C̃_{n,k}(j; X).

Proposition 5.8. We have the following natural stable homotopy equivalence.

F_{−q}C_{n,k}(X)/F_{−q−1}C_{n,k}(X) ≃_{S} F_{−q}C_{n}(X)/F_{−q−1}C_{n}(X) ∧ ξ_{n,k}(ΣX).

Proof. The proof is parallel to that of the stable homotopy equivalence

F_{−q}E_{n}(CX, X)/F_{−q−1}E_{n}(CX, X) ≃_{S} F_{−q}C_{n}(X)/F_{−q−1}C_{n}(X) ∧ C_{n−1}(ΣX)

in [12] and omitted.

Proof of Theorem 1.2. Let {E^{r}(C_{n,k}(X)), d^{r}} be the spectral sequence induced by applying a homology theory h_{∗}(−) to the stable filtration {F_{−q}C_{n,k}(X)} defined in Definition 5.7. The spectral sequence is a direct sum of small spectral sequences corresponding to the Snaith splitting and each summand has finite filtration. And 1 and 4 in Theorem 1.2 follow.

Since the filtration is defined in terms of little cubes, 2 is obvious.

Note that Proposition 5.8 together with 4 in Theorem 3.1 implies that, if h_{∗}(−) is multiplicative and both h_{∗}(ξ_{n,k}(ΣX)) and h_{∗}(C_{n−1}(X)) are h_{∗}-flat,

h̃_{∗}(F_{−q}C_{n,k}(X)/F_{−q−1}C_{n,k}(X)))

≅ h̃_{∗}(F_{−q}C_{n}(X)/F_{−q−1}C_{n}(X)) ⊗ h_{∗}(ξ_{n,k}(ΣX))
≅ h̃_{∗}(F_{−q}C_{n}(X)/F_{−q−1}C_{n}(X)) ⊗ (h_{∗}(C_{n−1}(ΣX))□_{h_{∗}(C_{n−1}(ΣX))}h_{∗}(ξ_{n,k}(ΣX)))
≅ (h̃_{∗}(F_{−q}C_{n}(X)/F_{−q−1}C_{n}(X)) ⊗ h_{∗}(C_{n−1}(ΣX))) □_{h_{∗}(C_{n−1}(ΣX))}h_{∗}(ξ_{n,k}(ΣX))
≅ h̃_{∗}(F_{−q}E_{n}(CX, X)/F_{−q−1}E_{n}(CX, X))□_{h_{∗}(C_{n−1}(ΣX))}h_{∗}(ξ_{n,k}(ΣX)).

This implies that the spectral sequence induced by the stable filtration on C_{n,k}(X) is the spectral sequence of Larry Smith defined by pulling back the (stable) injective display of E_{n}(CX, X) and 3 in Theorem 1.2 follows.

6 Applications

In this section, we apply the spectral sequences defined in the previous sections to compute K(n)_{∗}(Ω^{4}S^{2ℓ+1}). Let us assume everything is localized at an odd prime p in the following of this section.

We first need the Morava K-theory of Ω^{3}S^{2ℓ+1}. This can be computed by using the cobar spectral sequence

E^{2} ≅ Cotor^{K(n)_{∗}(Ω^{2}S^{2ℓ+1})}(K(n)_{∗}, K(n)_{∗}) ⟹ K(n)_{∗}(Ω^{3}S^{2ℓ+1}).

The nontrivial differentials can be determined by comparing with the first nontrivial Atiyah-Hirzebruch spectral sequence, i.e. v_{n}-Bockstein operation.

Proposition 6.1. We have the following isomorphism of algebras

K(n)_{∗}(Ω^{3}S^{2ℓ+1}) ≅ P (Q_{1}^{a}x|a ≤ n)
⊗E(Q_{1}^{b}βQ_{2}^{a+1}x|a + b ≤ n − 1)
⊗T P_{n−b}(βQ_{1}^{b}βQ_{2}^{a+1}x|1 ≤ b ≤ n − 1)

where x is the generator of K(n)_{∗}(S^{2ℓ−2}), deg Q_{i}y = p deg y + i(p − 1) and T P_{n−b} denotes the truncated polynomial algebra of height p^{n−b}.

The coalgebra structure of K(n)_{∗}(Ω^{3}S^{2ℓ+1}) can be determined as follows.

Lemma 6.2. Let V be the Verschiebung map on K(n)_{∗}(Ω^{3}S^{2ℓ+1}). Then we have the following.

V (v_{n}^{−1}βQ_{1}^{b}βQ_{2}^{a+1}x) = AQ_{0}^{b−1}βQ_{1}^{n−b}βQ_{2}^{a+b+1−n}x for a + b ≥ n (6)
V (v_{n}^{−1}Q_{2}^{n}x) = BQ_{0}^{n−1}x  (7)

for some A, B ∈ ℤ/pℤ^{×} and V is trivial on other generators.

Remark 6.3. Nonzero coefficients in K(n)_{∗}, like A, B and v_{n}^{−1} in the above formulas, do not make any difference in the following calculations. We will suppress those coefficients. For example, (6) and (7) will be refered to as

V (βQ_{1}^{b}βQ_{2}^{a+1}x) = Q_{0}^{b−1}βQ_{1}^{n−b}βQ_{2}^{a+b+1−n}x
V (Q_{2}^{n}x) = Q_{0}^{n−1}x.

Proof. Consider V (βQ_{1}^{b}βQ_{2}^{a+1}x) first. Since V commutes with the Frobenius map F, V (βQ_{1}^{b}βQ_{2}^{a+1}x) is of finite height and is a sum of elements of the form

Q_{0}^{c}βQ_{1}^{d}βQ_{2}^{e}x.

Since V divides the Snaith degree by p, we have

a + b = c + d + e.

By comparing the degrees, we have

p^{b} + 1 ≡ p^{c+d+1} + p^{c+1} mod p^{n} − 1.

Since b ≤ n − 1 and c < n − d, the only possibility is

b = c + 1
c + d + 1 = n

and

V (βQ_{1}^{b}βQ_{2}^{a+1}x) = Q_{0}^{b−1}βQ_{1}^{n−b}βQ_{2}^{a+b+1−n}x or 0.

To see V (βQ_{1}^{b}βQ_{2}^{a+1}x) is nontrivial, consider the mod p Hopf invariant

H_{p} : Ω^{3}S^{2ℓ+1} ⟶ Ω^{3}S^{2ℓp+1}.

In homology,

(H_{p})_{∗}(βQ_{1}^{b}βQ_{2}^{a+1}x) = βQ_{1}^{b}βQ_{2}^{a}x̄

where x̄ is the bottom cell of Ω^{3}S^{2ℓp+1}. Hence it is enough to prove

V (βQ_{1}^{b}βQ_{2}^{n−b+1}x) ≠ 0.

Suppose βQ_{1}^{b}βQ_{2}^{n−b+1}x is primitive. Note that from the above calculation it follows that βQ_{1}^{b}βQ_{2}^{a+1}x is primitive for a+b ≤ n, hence βQ_{1}^{n−b}βQ_{2}x is primitive. The E^{2}-term of the cobar EMSS for

Ω^{4}S^{2ℓ+1} ⟶ P Ω^{3}S^{2ℓ+1} ⟶ Ω^{3}S^{2ℓ+1}

contains

E(βQ_{2}^{b}βQ_{3}^{n−b+1}x′) ⊗ ⊗P (βQ_{1}^{b}βQ_{2}^{n−b}βQ_{3}x′).

However, we have the following formula

β_{v_{n}}^{1}(βQ_{2}^{b}βQ_{3}^{n−b+1}x′) = βQ_{1}^{b}βQ_{2}^{n−b}βQ_{3}x′.

(Proposition 2.24) for the first nontrivial differential in the Atiyah-Hirzebruch spectral sequence

E^{2} ≅ H_{∗}(Ω^{4}S^{2ℓ+1}; ℤ/pℤ) ⊗ K(n)_{∗} ⟹ K(n)_{∗}(Ω^{4}S^{2ℓ+1}).

Thus both βQ_{2}^{b}βQ_{3}^{n−b+1}x′ and βQ_{1}^{b}βQ_{2}^{n−b}βQ_{3}x′ must be killed by differentials in the cobar spectral sequence. Since

βQ_{2}^{b}βQ_{3}^{n−b+1}x′ ∈ E_{−1,∗}^{r}

this element must support a nontrivial differential. Since the cobar differentials respect the Snaith degrees, the only possibility is

d^{1}(βQ_{2}^{b}βQ_{3}^{n−b+1}x′) = βQ_{1}^{b}βQ_{2}^{n−b}βQ_{3}x′.

If we use the cobar construction to compute the E^{2}-term, then d^{1} is given by the Verschiebung map

d^{1}(z) = [V (z)|V (z)^{p−1}] = βQ_{1}(τ(V (z)))

and the result follows.

The case of V (Q_{2}^{n}x) is similar. It follows from the following formula in H_{∗}(Ω^{4}S^{2ℓ+1}; ℤ/pℤ)

β_{v_{n}}^{1}Q_{3}^{n}x = Q_{2}^{n−1}(−βQ_{1}x)
= −βQ_{1}^{n}x.

The second equality follows from the Adem relation for the homology operations. The details are omitted.

In the fibration

ΩS^{2ℓ+1} ⟶^{E^{2}} Ω^{3}S^{2ℓ+1} ⟶ W (ℓ),

K(n)_{∗}(ΩS^{2ℓ+1}) ≅ K(n)_{∗}[x] and the double suspension induces an injective map in K(n)_{∗}. Thus we obtain

Proposition 6.4. We have the following isomorphism of algebras

K(n)_{∗}(W (ℓ)) ≅ K(n)_{∗}(Ω^{3}S^{2ℓ+1})//K(n)_{∗}(ΩS^{2ℓ+1})
≅ P (Q_{1}^{a}x|1 ≤ a ≤ n)
⊗E(Q_{1}^{b}βQ_{2}^{a+1}x|a + b ≤ n − 1)
⊗T P_{n−b}(βQ_{1}^{b}βQ_{2}^{a+1}x|1 ≤ b ≤ n − 1).

The only nontrivial Verschiebung maps are

V (βQ_{1}^{b}βQ_{2}^{a+1}x) = Q_{0}^{b−1}βQ_{1}^{n−b}βQ_{2}^{a+b+1−n}x

for a + b ≥ n.

In order to compute K(n)_{∗}(Ω^{4}S^{2ℓ+1}), we first compute K(n)_{∗}(ΩW (ℓ)) by using the cobar spectral sequence for

ΩW (ℓ) ⟶ P W (ℓ) ⟶ W (ℓ).

Theorem 6.5. There exists an isomorphism of algebras

K(n)_{∗}(ΩW (ℓ)) ≅ E(Q_{1}^{b}Q_{3}^{a+1}x′|a + b ≤ n − 1)
⊗T P_{n}(βQ_{1}^{b}Q_{3}^{a+1}x′|a ≤ n)
⊗T P_{n−b−1}(Q_{2}^{b+1}βQ_{3}^{a+1}x′|a + b ≤ n − 2)
⊗E(Q_{1}^{c}βQ_{2}^{b+1}βQ_{3}^{a+1}x′|a + b + c ≤ n − 2)
⊗T P_{n−b−c−2}(βQ_{1}^{c+1}βQ_{2}^{b+1}βQ_{3}^{a+1}x′|b + c ≤ n − 3)

Proof. Consider the case n = 2 for simplicity. The calculation is exactly the same for n > 2.

We have

K(2)_{∗}(W (ℓ)) ≅ P (Q_{2}x, Q_{2}^{2}x)
⊗E(βQ_{2}x, βQ_{2}^{2}x)
⊗E(Q_{1}βQ_{2}x)
⊗T P_{1}(βQ_{1}βQ_{2}^{a+1}x).

and

V (βQ_{1}βQ_{2}^{a+1}x) = βQ_{1}βQ_{2}^{a}x

Thus

K(2)_{∗}(W (ℓ)) ≅ P (Q_{2}x, Q_{2}^{2}x)
⊗E(βQ_{2}x, βQ_{2}^{2}x)
⊗E(Q_{1}βQ_{2}x)
⊗Γ(βQ_{1}βQ_{2}x)

as Hopf algebras.

It is easy to compute the E^{2}-term of the cobar EMSS

Cotor^{K(2)_{∗}(W (ℓ))}(K(n)_{∗}, K(n)_{∗}) ≅ E(Q_{1}^{a}x′, Q_{1}^{a}Q_{3}^{2}x′|a ≥ 0)

⊗P (βQ_{1}^{a+1}Q_{3}x′, βQ_{1}^{a+1}Q_{3}^{2}x′|a ≥ 0)

⊗P (βQ_{3}x′, βQ_{3}^{2}x′)
⊗P (Q_{2}βQ_{3}x′) ⊗ E(βQ_{2}βQ_{3}x′)
≅ E(Q_{1}^{a}x′, Q_{1}^{a}Q_{3}^{2}x′|a ≥ 0)
⊗P (βQ_{1}^{a}Q_{3}x′, βQ_{1}^{a}Q_{3}^{2}x′|a ≥ 0)
⊗P (Q_{2}βQ_{3}x′) ⊗ E(βQ_{2}βQ_{3}x′).

By comparing with the Atiyah-Hizrbuch differentials and by counting three kinds of degrees (the Snaith degree, the total degree and the filtration degree), we have the following differentials

d^{2p^{2}−1}Q_{1}^{a}Q_{3}x′ = \begin{cases}Q_{0}^{2}βQ_{1}^{a−2}Q_{3}x′ & a ≥ 2 \\ 0 & a < 2 \end{cases}

d^{r}Q_{1}^{a}Q_{3}^{2}x′ = \begin{cases}Q_{0}^{2}βQ_{1}^{a−2}Q_{3}^{2}x′ & a ≥ 2 (r = 2p^{2} − 1) \\ Q_{0}βQ_{2}βQ_{3}x′ & a = 1 (r = p − 1) \\ 0 & a = 0 \end{cases}

and we don’t have any higher differentials. And thus

E^{∞} ≅ E(Q_{3}x′, Q_{3}^{2}x′, Q_{1}Q_{3}x′)
⊗T P_{2}(βQ_{1}^{a}Q_{3}x′)
⊗T P_{2}(βQ_{1}^{a}Q_{3}^{2}x′)
⊗T P_{1}(βQ_{2}βQ_{3}x′)
⊗E(Q_{2}βQ_{3}x′).

It is elementary to check that there is no nontrivial algebra extensions in E^{∞}-term by counting degrees and the result follows.

The above result and the cobar spectral sequence for the fibration

Ω^{4}S^{2ℓ+1} ⟶ ΩW (ℓ) ⟶^{Ωπ} ΩS^{2ℓ+1}.

gives us K(n)_{∗}(Ω^{4}S^{2ℓ+1}).

Theorem 6.6. We have the following isomorphism of K(n)_{∗}-modules

K(n)_{∗}(Ω^{4}S^{2ℓ+1}) ≅ E(Q_{1}^{a}x′|a ≤ 2n)
⊗T P_{n−a−1}(βQ_{1}^{a+1}x′|a ≤ n − 2)
⊗T P_{n}(βQ_{1}^{a+n+1}x′|a ≥ 0)
⊗E(Q_{1}^{b}Q_{3}^{a}x′|a + b ≤ n, a ≥ 1, b ≥ 1)
⊗T P_{n}(βQ_{1}^{b}Q_{3}^{a}x′|1 ≤ a ≤ n)
⊗T P_{n−b−1}(Q_{2}^{b+1}βQ_{3}^{a+1}x′|a + b ≤ n − 2)
⊗E(Q_{1}^{c}βQ_{2}^{b+1}βQ_{3}^{a+1}x′|a + b + c ≤ n − 2)

⊗T P_{n−b−c−2}(βQ_{1}^{c+1}βQ_{2}^{b+1}βQ_{3}^{a+1}x′|b + c ≤ n − 3)

Proof. By the above calculation, there is no element of infinite height in K(n)_{∗}(ΩW (ℓ)). Thus

(Ωπ)_{∗} : K(n)_{∗}(ΩW (ℓ)) ⟶ K(n)_{∗}(ΩS^{2ℓ+1})

is trivial and it is easy to determine the E^{2}-term of the cobar EMSS.

E^{2} ≅ Cotor^{K(n)_{∗}(ΩS^{2ℓ+1})}(K(n)_{∗}, K(n)_{∗}) ⊗ K(n)_{∗}(ΩW (ℓ))
≅ E(Q_{1}^{a}x′|a ≥ 0) ⊗ P (βQ_{1}^{a+1}x′|a ≥ 0)
⊗E(Q_{1}^{b}Q_{3}^{a+1}x′|a + b ≤ n − 1)
⊗T P_{n}(βQ_{1}^{b}Q_{3}^{a+1}x′|a ≤ n)
⊗T P_{n−b−1}(Q_{2}^{b+1}βQ_{3}^{a+1}x′|a + b ≤ n − 2)
⊗E(Q_{1}^{c}βQ_{2}^{b+1}βQ_{3}^{a+1}x′|a + b + c ≤ n − 2)
⊗T P_{n−b−c−2}(βQ_{1}^{c+1}βQ_{2}^{b+1}βQ_{3}^{a+1}x′|b + c ≤ n − 3).

In order to determine differentials, we again use the Atiyah-Hirzebruch differential. The followings hold in H_{∗}(Ω^{4}S^{2ℓ+1}; ℤ/pℤ)

β_{v_{n}}^{1}Q_{3}^{n}x′ = −βQ_{1}^{n}x′
β_{v_{n}}^{1}Q_{1}^{a}Q_{3}^{n−a}x′ = −Q_{0}^{a}βQ_{1}^{n−a}x

and we have

d^{2}Q_{3}^{n}x′ = −βQ_{1}^{n}x′
d^{2p^{a}−1}Q_{1}^{a}Q_{3}^{n−a}x′ = −Q_{0}^{a}βQ_{1}^{n−a}x

and

E^{2p^{n−1}}  ≅ E(Q_{1}^{a}x′|a ≥ 0)
⊗T P_{n−a}(βQ_{1}^{a}x′|1 ≤ a ≤ n − 1)
⊗P (βQ_{1}^{a+n+1}x′|a ≥ 0)
⊗E(Q_{1}^{b}Q_{3}^{a}x′|a + b ≤ n, a ≥ 1, b ≥ 1)
⊗T P_{n}(βQ_{1}^{b}Q_{3}^{a+1}x′|a ≤ n)
⊗T P_{n−b−1}(Q_{2}^{b+1}βQ_{3}^{a+1}x′|a + b ≤ n − 2)
⊗E(Q_{1}^{c}βQ_{2}^{b+1}βQ_{3}^{a+1}x′|a + b + c ≤ n − 2)
⊗T P_{n−b−c−2}(βQ_{1}^{c+1}βQ_{2}^{b+1}βQ_{3}^{a+1}x′|b + c ≤ n − 3).

The following map of fibrations

[figure: Ω^{2}S^{2ℓ+1}; P ΩS^{2ℓ+1}; ΩS^{2ℓ+1}; Ω^{4}S^{2ℓ+1}; ΩW (ℓ); ΩS^{2ℓ+1}; Ω^{2}E^{2}; ∥; Ωπ]

induces a map of cobar EMSSes. The cobar EMSS for the upper fibration is nothing but a reindexed Atiyah-Hirzebruch spectral sequence for K(n)_{∗}(Ω^{2}S^{2ℓ+1}) and the differentials are given by

d^{2p^{n}−1}(Q_{1}^{a}x) = Q_{0}^{n}βQ_{1}^{a−n}x.

Hence we have

d^{2p^{n}−1}(Q_{1}^{a}x′) = \begin{cases}Q_{0}^{n}Q_{1}^{a−n}x′ & a ≥ 2n + 1 \\ 0 & a ≤ 2n \end{cases}

in our cobar EMSS.

It is tedious but elementary to check that there is no higher differentials by counting degrees. And we have

E^{∞} ≅ E(Q_{1}^{a}x′|a ≤ 2n)
⊗T P_{n−a−1}(βQ_{1}^{a+1}x′|a ≤ n − 2)
⊗T P_{n}(βQ_{1}^{a+n+1}x′|a ≥ 0)
⊗E(Q_{1}^{b}Q_{3}^{a}x′|a + b ≤ n, a ≥ 1, b ≥ 1)
⊗T P_{n}(βQ_{1}^{b}Q_{3}^{a}x′|1 ≤ a ≤ n)
⊗T P_{n−b−1}(Q_{2}^{b+1}βQ_{3}^{a+1}x′|a + b ≤ n − 2)
⊗E(Q_{1}^{c}βQ_{2}^{b+1}βQ_{3}^{a+1}x′|a + b + c ≤ n − 2)
⊗T P_{n−b−c−2}(βQ_{1}^{c+1}βQ_{2}^{b+1}βQ_{3}^{a+1}x′|b + c ≤ n − 3)

Remark 6.7. We do have nontrivial algebra extensions in the above E^{∞}-term. For example, when n = 2 we have

(Q_{2}βQ_{3}x′)^{p} = βQ_{1}^{3}x′
(βQ_{1}^{a}Q_{3}^{2}x′)^{p^{2}} = βQ_{1}^{a+4}x′

and

K(2)_{∗}(Ω^{4}S^{2ℓ+1}) ≅ E(Q_{1}^{a}x′|0 ≤ a ≤ 4)
⊗T P_{1}(βQ_{1}x′)
⊗T P_{2}(βQ_{1}^{a}Q_{3}x′|a ≥ 0)
⊗T P_{3}(Q_{2}βQ_{3}x′)
⊗T P_{4}(βQ_{1}^{a}Q_{3}^{2}x′|a ≥ 0)
⊗E(βQ_{2}βQ_{3}x′)

as algebras.

7 Concluding Remarks

Note that the only nontrivial data we have used in our calculation of K(n)_{∗}(Ω^{4}S^{2ℓ+1}) in the previous section is the Nishida relation for the v_{n}-Bockstein operation (Proposition 2.24). The formula determines the coalgebra structure and the differentials.

On the other hand, as we stated in §1, Langsetmo’s result on K(1)_{∗}(Ω^{k}S^{2ℓ+1}) implies that in the K(1)-cobar EMSS for

Ω^{4}S^{2ℓ+1} ⟶ P Ω^{3}S^{2ℓ+1} ⟶ Ω^{3}S^{2ℓ+1}

we have higher differentials which cannot be determined by the v_{1}-Bockstein operation directly.

This simplicity of the cobar EMSSes related to W (ℓ) is closely related to the fact that the secondary suspension

σ : W (ℓ) ⟶ Ω^{2p}W (ℓ + 1)

induces an isomorphism in K(1). In fact, we can easily continue the calculations in Theorem 6.5 to get K(2)_{∗}(Ω^{k}W (ℓ)) for k ≤ 2p.

Theorem 7.1. For 0 ≤ k ≤ p − 1, we have

K(2)_{∗}(Ω^{2k−1}W (ℓ)) ≅ E(Q_{2k+1}x) ⊗ T P_{2}(βQ_{2k+1}x)
⊗E(Q_{2i−1}Q_{2k+1}x|1 ≤ i ≤ k + 1)
⊗T P_{2}(βQ_{1}^{a}Q_{2i−1}Q_{2k+1}x|1 ≤ i ≤ k + 1)
⊗T P_{1}(Q_{2i}βQ_{2k+1}x|1 ≤ i ≤ k)
⊗E(βQ_{2i}βQ_{2k+1}x|1 ≤ i ≤ k)

as algebras and the coalgebra structure is given by

V (βQ_{1}Q_{2i+1}Q_{2k+1}x) = Q_{2i}βQ_{2k+1}x
V (βQ_{1}^{a+2}Q_{2i+1}Q_{2k+1}x) = Q_{0}βQ_{1}^{a}Q_{2i+1}Q_{2k+1}x
V (βQ_{1}^{2}Q_{2k+1}x) = Q_{0}βQ_{2k+1}x.

Theorem 7.2. For 0 ≤ k ≤ p − 1, we have

K(2)_{∗}(Ω^{2k}W (ℓ)) ≅ P (Q_{2k+2}x) ⊗ E(βQ_{2k+2}x)
⊗P (Q_{2i}Q_{2k+2}x|1 ≤ i ≤ k + 1)
⊗E(βQ_{2i}Q_{2k+2}x|1 ≤ i ≤ k + 1)
⊗E(Q_{2i−1}βQ_{2k+2}x|1 ≤ i ≤ k + 1)
⊗Γ(βQ_{2i−1}βQ_{2k+2}x|1 ≤ i ≤ k + 1)

as Hopf algebras.

Theorem 7.3. We have

K(2)_{∗}(Ω^{2p−1}W (ℓ)) ≅ E(Q_{2p+1}x) ⊗ T P_{2}(βQ_{2p+1}x)
⊗E(Q_{1}^{2}Q_{2p+1}x) ⊗ T P_{2}(βQ_{1}^{a}Q_{2p+1}^{2}x)
⊗E(Q_{2i+1}Q_{2p+1}x|0 ≤ i ≤ p − 1)
⊗T P_{2}(βQ_{1}^{a}Q_{2i+1}Q_{2p+1}x|0 ≤ i ≤ p − 1)
⊗T P_{1}(Q_{2i}βQ_{2p+1}x|1 ≤ i ≤ p)
⊗E(βQ_{2i}βQ_{2p+1}x|1 ≤ i ≤ p)

as algebras. The coalgebra structure is given by the corresponding formulas in Theorem 7.1 together with

V (βQ_{2p+1}^{2}x) = βQ_{2p+1}x.

Theorem 7.4. We have

K(2)_{∗}(Ω^{2p}W (ℓ)) ≅ P (Q_{2p+2}x) ⊗ E(βQ_{2p+2}x)
⊗P (Q_{2}^{2}Q_{2p+2}x) ⊗ E(βQ_{2}^{2}Q_{2p+2}x)
⊗P (Q_{2i+2}Q_{2p+2}x|0 ≤ i ≤ p − 1)
⊗E(βQ_{2i+2}Q_{2p+2}x|0 ≤ i ≤ p − 1)
⊗E(Q_{2i+1}βQ_{2p+2}x|1 ≤ i ≤ p)
⊗Γ(βQ_{2i+1}βQ_{2p+2}x|1 ≤ i ≤ p)
⊗E(Q_{1}βQ_{2}Q_{2p+2}x)
⊗Γ(βQ_{1}βQ_{2}Q_{2p+2}x)

as Hopf algebras.

Up to this stage, the first nontrivial Atiyah-Hirzebruch differential, i.e. v_{2}- Bockstein operation determines everything. However nontrivial higher differentials appear in the K(2)-cobar EMSS for

Ω^{2p+1}W (ℓ) ⟶ P Ω^{2p}W (ℓ) ⟶ Ω^{2p}W (ℓ).

Thus computing K(2)_{∗}(Ω^{2p+1}W (ℓ)) by the cobar EMSS for the path-loop fibration seems to be as difficult as computing K(1)_{∗}(Ω^{4}S^{2ℓ+1}) by using the cobar EMSS for the path-loop fibration.

We also note that computing K(1)_{∗}(Ω^{k}W (ℓ)) by using the cobar EMSS for the path-loop fibration

Ω^{k}W (ℓ) ⟶ P Ω^{k−1}W (ℓ) ⟶ Ω^{k−1}W (ℓ)

is straightforward and is as easy as computing K(0)_{∗}(Ω^{k}S^{2ℓ+1}) ≅ H_{∗}(Ω^{k}S^{2ℓ+1}; ℚ). Let W^{(2)}(ℓ) be the fiber of the secondary suspension map

W (ℓ) ⟶ Ω^{2p}W (ℓ + 1).

These facts suggests that computing K(2)_{∗}(Ω^{k}W^{(2)}(ℓ)) would be as easy as computing K(1)_{∗}(Ω^{k}W (ℓ)), if the cobar EMSS for the path-loop fibration on Ω^{k}W^{(2)}(ℓ) were to converge.

In fact, a recent work of Arone’s [1] suggests that W (ℓ) and its higher analogues defined by Weiss in [16] are useful tools for studying “v_{n}-periodic part” of S^{2ℓ−1}.

Theorem 7.5 (Weiss, Arone). There exists a natural tower of fibrations

[figure: ⋮; W^{(2)}(X); Ω^{6}W^{(2)}(Σ^{2}X); W^{(1)}(X); Ω^{4}W^{(1)}(Σ^{2}X); X; Ω^{2}Σ^{2}X.; w_{3}; w_{2}; E^{2}]

with the following property: localized at an odd prime p,

w_{p^{n}} : W^{(p^{n}−1)}(S^{2ℓ−1}) ⟶ Ω^{2p^{n}}W^{(p^{n}−1)}(S^{2ℓ+1})

is a v_{n}-periodic homotopy equivalence.

This result suggests that computing K(n)_{∗}(Ω^{k}W^{(p^{n}−1)(S^{2ℓ+1})}) for all k would be tractable and we would have a systematic way of computing K(n)_{∗}(Ω^{k}S^{2ℓ+1}), if the following conjecture were to be true.

Conjecture 7.6. The cobar EMSS for the fibrations

Ω^{k}W^{(m)}(X) ⟶ P Ω^{k−1}W^{(m)}(X) ⟶ Ω^{k−1}W^{(m)}(X)
Ω^{2m+1+k}W^{(m)}(Σ^{2}X) ⟶ Ω^{k}W^{(m)}(X) ⟶ Ω^{k}W^{(m−1)}(X)

converge strongly for any homology theory.

References

[1] G. Arone. Iterates of the suspension map and Mitchell’s finite spectra with A_{k}-free cohomology. Preprint, 1998.

[2] F.R. Cohen. The unstable decomposition of Ω^{2}Σ^{2}X and its applications. Mathematische Zeitschrift, 182:553–568, 1983.

[3] A. Dold and R. Thom. Quasifaserungen und unendliche symmetrische produkte. Annals of Mathematics, 67:239–281, 1958.

[4] I.M. James. Reduced product spaces. Annals of Mathematics, 62:170–197, 1955.

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