Morfismos, Vol. XX, No. XX, 199X, pp. X–X

LITTLE CUBES AND HOMOTOPY THEORY ^{∗}

DAI TAMAKI ^{1}

Abstract

This is a survey article on the theory of configuration space of little cubes, which is naturally related to the homotopy commutativity of double or more highly iterated loop spaces. The article begins with historical background and then surveys important applications to homotopy theory.

1991 Mathematics Subject Classification: 55-02, 55P35 Keywords and phrases: Configuration space, loop space, quasifibration

1 Introduction

A little n-cube is the image of an Affine embedding

c : I^{n} ↪ I^{n}

whose image has all edges parallel to the corresponding edges of the standard n-cube I^{n}.

∗Invited article 1 Department of Mathematical Sciences, Faculty of Science, Shinshu University, Japan.

c

The configuration space of j little n-cubes C_{n}(j) is the set of j-tuples (c_{1}, ⋯, c_{j}) of little n-cubes whose images have disjoint interiors from each other. C_{n}(j) is topologized with the compact-open topology.

Since a little cube is determined by its image, it is safe to discuss on little cubes by drawing a picture like:

c_{1}  c_{2}

The reader might notice that he or she has seen this kind of picture in a textbook of algebraic topology.

It is an elementary theorem in homotopy theory that the n-th homotopy group of a pointed space X, π_{n}(X), is an Abelian group, or equivalently the n-th loop space on X, Ω^{n}X, is a homotopy commutative H-space, if n ≥ 2.

In most textbooks on algebraic topology, this fact is explained as follows: Suppose an element α ∈ π_{n}(X) is represented by a continuous map

f : (I^{n}, ∂I^{n}) ⟶ (X, ∗),

LITTLE CUBES AND HOMOTOPY THEORY 3

where ∗ denotes the base point of X. Pick up another element β ∈ π_{n}(X) represented by

g : (I^{n}, ∂I^{n}) ⟶ (X, ∗).

The product of α and β in π_{n}(X), α + β, is represented by the map

f + g : (I^{n}, ∂I^{n}) ⟶ (X, ∗)

defined by the following picture.

[figure: ^{...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................}; X; ∗; ^{........................................................................................................................................................................................................................................................................................................................................}; ^{.........................................................................................................................................................................................................................................................................................................................................................................................}; f; ^{...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................}; g; ^{...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................}]

On the other hand, β + α is represented by the map

g + f : (I^{n}, ∂I^{n}) ⟶ (X, ∗)

defined by the following picture.

[figure: ^{...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................}; X; ∗; ^{........................................................................................................................................................................................................................................................................................................................................}; ^{.........................................................................................................................................................................................................................................................................................................................................................................................}; g; ^{..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................}; f; ^{.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................}]

If n ≥ 2, a homotopy connecting these two maps can be obtained by rotating two (rectangular) cubes in I^{n}.

[figure: f; g; f; g; g; f; g; f]

Notice that domains of f + g and g + f can be regarded as elements of C_{n}(2) and the above picture can be interpreted as a path in C_{n}(2), since we need to keep two cubes separated while rotating.

In fact, C_{n}(2) is path-connected (more generally C_{n}(j) is known to be (n − 2)-connected). The path-connectivity of C_{n}(2) implies the commutativity of π_{n}(X) or the homotopy commutativity of Ω^{n}X. This can be considered as the first application of the topology of little cubes to homotopy theory.

It is natural to extend this idea to study higher homotopy commutativity of iterated loop spaces. Boardman and Vogt began the first systematic study in this direction [5, 6]. A few years later, Peter May established a concrete relationship between iterated loop spaces and little cubes by proving the recognition principle [40]. The study of little cubes also led him to introduce the notion of operad, which turned out to be an important object naturally appearing in many fields of mathematics and mathematical physics. But we are not going to cover operad in this article. Those who are interested in operad are recommended to take a look at [36], for example.

The following is the organization of this article:

Section 2. Basic Definitions and Fundamental Facts: recalls basic definitions and classical results used in later sections.

Section 3. Stable Splitting of Iterated Loop Spaces: reviews the development of homotopy theory stimulated by Snaith’s theorem of stable splitting of Ω^{n}Σ^{n}X.

Section 4. Constructing Maps and Spaces: shows that various important maps and spaces have been constructed by using little cubes.

Section 5. Problems: is a collection of open problems related to little cubes.

2 Basic Definitions and Fundamental Facts

Let us begin with a precise definition of little cube.

Definition 2.1.1 For a positive integer n, a little n-cube c is a map I^{n} ⟶ I^{n} which can be decomposed into the following form:

c = ℓ_{1} × ⋯ × ℓ_{n}

where each ℓ_{i} : I ⟶ I is an orientation preserving Affine embedding. The space of little n-cubes is denoted by C_{n}(1) and topologized with compact-open topology. Thus C_{n}(1) is a subspace of Map(I^{n}, I^{n}).

The configuration space of j little n-cubes C_{n}(j) is defined as follows:

C_{n}(j) = {(c_{1}, ⋯, c_{j}) ∈ C_{n}(1)^{j}|c_{i}(IntI^{n}) ∩ c_{k}(IntI^{n}) = ∅ if i ≠ k}

C_{n}(j) admits a natural action of the symmetric group of j-letters, Σ_{j}, by permuting the indices of little cubes.

C_{n} = {C_{n}(j)}_{j} is called the little n-cube operad.

We also need the case n = ∞ to study infinite loop spaces. Define

C_{∞}(j) = colim_{\overrightarrow{n}} C_{n}(j)

where colimit is taken over the inclusion maps induced by

I^{n} × {0} ↪ I^{n+1}.

Obviously C_{∞}(j) inherits the action of Σ_{j}.

Thanks to the following fact, we do not have to struggle with the compact-open topology on C_{n}(j).

Lemma 2.1.2 Define a map

ξ_{n} : C_{n}(1) ⟶ I^{2n}

by ξ_{n}(c) = (c(\frac{1}{4}, ⋯, \frac{1}{4}), c(\frac{3}{4}, ⋯, \frac{3}{4})), then ξ_{n} is an embedding of C_{n}(1) as an open subset of I^{2n}.

As is stated in Introduction, the following is one of the most fundamental properties of little cube.

Lemma 2.1.3 C_{n}(j) is (n − 2)-connected. Hence C_{∞}(j) is contractible.

The following map plays a central role in studying the relationship between the configuration space of little cubes C_{n}(j) and an n-fold loop space Ω^{n}X.

Definition 2.1.4 For a pointed space X, define

θ_{n,j} : C_{n}(j) × (Ω^{n}X)^{j} ⟶ Ω^{n}X

as follows: For (c_{1}, ⋯, c_{j}) ∈ C_{n}(j) and ω_{1}, ⋯, ω_{j} ∈ Ω^{n}X,

θ_{n,j}(c_{1}, ⋯, c_{j}; ω_{1}, ⋯, ω_{j}) : I^{n} ⟶ X

is defined to be ω_{i} on each cube c_{i}(I^{n}) and maps the outside of little cubes to the basepoint. More precisely

θ_{n,j}(c_{1}, ⋯, c_{j}; ω_{1}, ⋯, ω_{j})(t) =

= \{ \begin{matrix}ω_{i}(c_{i}^{−1}(t)) & if t ∈ c_{i}(I^{n}) for some 0 ≤ i ≤ j, \\ ∗ & otherwise. \end{matrix}

θ_{n,j} is obviously Σ_{j}-equivariant, hence we have an induced map on the quotient, which is denoted by the same notation:

θ_{n,j} : C_{n}(j) ×_{Σ_{j}} (Ω^{n}X)^{j} ⟶ Ω^{n}X

These maps {θ_{n,j}} satisfy the following compatibility conditions.

Lemma 2.1.5 Let ∗ denote the constant loop to the basepoint of Ω^{n}X, then, for (c_{1}, ⋯, c_{j}) ∈ C_{n}(j) and ω_{1}, ⋯, ω_{i−1}, ω_{i+1}, ⋯, ω_{j} ∈ Ω^{n}X, we have

θ_{n,j}(c_{1}, ⋯, c_{j}; ω_{1}, ⋯, ω_{i−1}, ∗, ω_{i+1}, ⋯, ω_{j})
= θ_{n,j−1}(c_{1}, ⋯, c_{i−1}, c_{i+1}, ⋯, c_{j}; ω_{1}, ⋯, ω_{i−1}, ω_{i+1}, ⋯, ω_{j})

Definition 2.1.6 For a pointed space Y, an action of C_{n} is a collection of maps

θ_{j} : C_{n}(j) ×_{Σ_{j}} Y^{j} ⟶ Y

for j = 0, 1, ⋯ satisfying the same relations as in Lemma 2.1.5.

In other words, an n-fold loop space has an action of C_{n}. May proved that, conversely, the existence of an action of C_{n} on a pointed space Y can be used as a criterion for Y to be equivalent to an n-fold loop space.

Theorem 2.1.7 (Recognition Principle) Let X be a path-connected space with a nondegenerate basepoint. Then X has a weak homotopy type of an n-fold loop space if and only if it has an action of little ncube operad.

The relations in Lemma 2.1.5 can be used to pierce the spaces C_{n}(j) ×_{Σ_{j}} (Ω^{n}X)^{j} together to get a single space C_{n}(Ω^{n}X) as follows.

Definition 2.1.8 Let Y be a pointed space with basepoint ∗. Generate an equivalence relation ∼ on ∐_{j} C_{n}(j) ×_{Σ_{j}} Y^{j} by relations

(c_{1}, ⋯, c_{j}; y_{1}, ⋯, y_{i−1}, ∗, y_{i+1}, ⋯, y_{j})
∼ (c_{1}, ⋯, c_{i−1}, c_{i+1}, ⋯, c_{j}; y_{1}, ⋯, y_{i−1}, y_{i+1}, ⋯, y_{j}).

Define C_{n}(Y) by

C_{n}(Y) = (∐_{j} C_{n}(j) ×_{Σ_{j}} Y^{j}) / ∼ .

Corollary 2.1.9 {θ_{n,j}} induces a well-defined continuous map

θ_{n} : C_{n}(Ω^{n}X) ⟶ Ω^{n}X.

Note that C_{n}(Y) can be defined for any pointed space Y and we have a natural map

σ_{n} : Y ⟶ C_{n}(Y)

induced by the inclusion of C_{n}(1) × Y. The reader can easily check that θ_{n} ∘ σ_{n} ≃ id if Y = Ω^{n}X. In other words, if Y is equivalent to an n-fold loop space Ω^{n}X, Y is a retract of C_{n}(Y).

[figure: Y; C_{n}(Y); Y.; σ_{n}; id; θ_{n}]

Notice that the evaluation map on Ω^{n}X

eval : Σ^{n}(Ω^{n}X) ⟶ X

and the Freudenthal suspension

E^{n} : X ⟶ Ω^{n}Σ^{n}X

make the following diagram commutative

[figure: Ω^{n}X; Ω^{n}Σ^{n}(Ω^{n}X); Ω^{n}X.; E^{n}; id; Ω^{n}eval]

This similarity between the functors C_{n} and Ω^{n}Σ^{n} can be explained by the following important theorem of May.

Theorem 2.1.10 (Approximation Theorem) Let X denote a pathconnected space with a nondegenerate basepoint. Then we have a natural weak homotopy equivalence

C_{n}(X) ≃ Ω^{n}Σ^{n}X.

Recognition Principle is important because it is the origin of the theory of operad. But for practical applications in homotopy theory, Approximation Theorem is far more useful, because it gives us a combinatorial model for iterated loop spaces and thus making them easier to handle.

Besides May’s model, some other combinatorial models for Ω^{n}Σ^{n}X have been discovered. The oldest is the reduced product of James [27] which is a model for ΩΣX. Milgram [41] constructed a model for Ω^{n}Σ^{n}X generalizing James’ construction. Segal [43] suggested to use the configuration space of distinct points instead of little cubes. The resulting space is equivalent to the little cube model. May’s approach was improved by J. Caruso and S. Waner [11] to prove an approximation theorem for nonconnected spaces. There are also simplicial models found by Milnor [42] for n = 1, by Barratt and Eccles [2] for n = ∞ and by Jeff Smith [44] for general n.

These are essentially equivalent to each other. Among these models, however, the “little cube model” (and its variants, like the one using configuration space of distinct points) has been most popular. This is partly because Victor Snaith used the little cube model to prove the stable splitting theorem, which is the subject of the next section.

3 Stable Splitting of Iterated Loop Spaces

3.1 Snaith’s Theorem

A stable splitting (in fact, a splitting after a single suspension) of ΩΣX was proved by James in 1955 [27] using a combinatorial construction of ΩΣX, called the James construction or reduced product. Important tools in classical homotopy theory, like the (James)-Hopf invariant

H : ΩS^{n+1} ⟶ ΩS^{2n+1}

and EHP fibration

S^{n} ⟶^{E} ΩS^{n+1} ⟶^{H} ΩS^{2n+1}

have been constructed by using this splitting.

The little cube model of Ω^{n}Σ^{n}X can be regarded as a generalization of the James construction. In fact, Snaith [45] proved a stable splitting theorem for Ω^{n}Σ^{n}X by using the little cube model.

Theorem 3.1.1 For a pointed space X with a nondegenerate basepoint and a positive integer n, we have a weak stable homotopy equivalence

Ω^{n}Σ^{n}X ≃_{S} ⋁_{j=1}^{∞} C_{n}(j)_{+} ∧_{Σ_{j}} X^{∧j}. (1)

This holds also for n = ∞.

Note that C_{n}(X) is naturally filtered by the number of cubes

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

and the right hand side of the above splitting is the “subquotient” of C_{n}(X) with respect to this filtration.

Later, Cohen, May and Taylor generalized Snaith’s theorem to more general settings [15]. For those who are curious about how to construct these stable splittings, the appendix of [12] will be of interest.

It is also worthwhile to note that these splittings can be obtained by a very simple argument in the category of spectra. This fact, proved by R.L. Cohen [24], depends on the existence of the Σ_{j}-equivariant half smash product of spectra constructed by Lewis, May and Steinberger

3.2 Applications of Stable Splitting

After the stable splitting theorem was proved by Snaith, the case n = 2 and X = S^{2k−1} became a subject of intense study in 70’s. Mahowald [38] found a new infinite family in the 2-primary components of the stable homotopy groups of spheres by using the summands of the stable splitting of Ω^{2}S^{9}. An odd primary analogue was obtained by R.L. Cohen [23].

These facts suggest that the cohomology of the summands of the Snaith splitting of Ω^{2}S^{2k+1} is equipped with an important module structure over the Steenrod algebra. To be more precise, we need the following definition.

Definition 3.2.1 For any prime p and integer k ≥ 0, define a module M(k, p) over the mod p Steenrod algebra as follows:

M(k, p) = \{ \begin{matrix}A_{2}/(χ(Sq^{i})|i > k) & if p = 2, \\ A_{p}/(χ(β^{ϵ}P^{i})|i > k, ϵ = 0, 1) & if p is odd, \end{matrix}

where χ denotes the canonical anti-automorphism on A_{p}.

Theorem 3.2.2 For any prime p and integer k ≥ 0, there exists a p-local spectrum B(k, p) satisfying the following properties:

1. H^{∗}(B(k, p); Z/pZ) ≅ M(k, p) as modules over the Steenrod algebra.

2. Let j_{k} : B(k, p) ⟶ HZ/pZ be a generator of H^{0}(B(k, p); Z/pZ). Then for any CW complex X, the induced map of generalized homology theories

(j_{k})_{∗} : B(k, p)_{∗}(X) ⟶ H_{∗}(X; Z/pZ)

is surjective for q ≤ 2k + 1 if p = 2 and for q ≤ 2p(k + 1) − 1 if p is odd.

Furthermore such a spectrum is unique up to homotopy.

Definition 3.2.3 B(k, p) is called the Brown-Gitler spectrum [7].

Mahowald in [38] conjectured that the stable summands of Ω^{2}S^{2k+1} localized at 2 realize the Brown-Gitler spectra. This fact was proved by Brown and Peterson [8].

Theorem 3.2.4 Localized at 2, we have the following homotopy equivalence

C_{2}(j)_{+} ∧_{Σ_{j}} (S^{2k−1})^{∧j} ≃ Σ^{j(2k−1)}B([\frac{j}{2}]).

Odd primary analogs are proved by R. Cohen [23].

Theorem 3.2.5 Localized at an odd prime p, we have the following homotopy equivalence when j = mp + r for p > r > 0:

C_{2}(pj)_{+} ∧_{Σ_{pj}} (S^{2k−1})^{∧pj} ≃ Σ^{j(2pk−2)}B(m).

There is another way of describing the summands of the stable splitting of Ω^{n}S^{n+k} closely related to the above theorems.

Consider the vector bundle

p_{n,j} : C_{n}(j) ×_{Σ_{j}} R^{j} ⟶ C_{n}(j)/Σ_{j}.

It is easy to see that the Thom complex of p_{n,j}, T (p_{n,j}), is C_{n}(j)_{+}∧_{Σ_{j}} S^{j} which is a stable summand of Ω^{n}S^{n+1}. More generally, we have

T (\underbrace{p_{n,j} ⊕ ⋯ ⊕ p_{n,j}}_{k}) = C_{n}(j)_{+} ∧_{Σ_{j}} S^{jk}.

It is worthwhile to note that the infinite families in the stable homotopy groups found by Mahowald and R. Cohen comes from the triviality of p_{2,j}⊕p_{2,j} proved by F. Cohen, Mahowald and Milgram [14]. The order of p_{n,j} in general was studied in [13].

Mahowald’s construction of infinite families in the stable homotopy groups of spheres is one of the most important applications of Ω^{2}S^{2k+1} in stable homotopy theory. Note that the classical applications of the James splitting of ΩS^{2k+1} live in the unstable world: For example, the mod p Hopf invariant

H_{p} : ΩS^{2n+1} ⟶ ΩS^{2np+1}

is defined to be the adjoint to the following composition

ΣΩS^{2n+1} ≃ Σ (⋁_{j}(S^{2n})^{∧j}) ⟶ Σ(S^{2n})^{∧p} = S^{2np+1}.

In order to get an unstable application of the splitting of Ω^{2}S^{2k+1}, for example a “secondary Hopf invariant”, we need to desuspend the Snaith splitting. However, as stated in [12], Ω^{2}Σ^{2}X does not split in finitely many suspensions. This is one of the crucial differences between ΩΣX and Ω^{n}Σ^{n}X for n ≥ 2.

This is the difficulty in applying Ω^{n}Σ^{n}X in unstable homotopy theory when n ≥ 2. But F. Cohen, May and Taylor [15] proved that each piece can be split off in finitely many suspensions.

Theorem 3.2.6 Fix a positive integer k, then there exists a positive integer L (depending on k) and a map

Σ^{L}C_{n}(X) ⟶ Σ^{L}C_{n}(k)_{+} ∧_{Σ_{k}} X^{∧k}

which is equivalent to the composition of the Snaith splitting and the projection on the k-th component after suspending infinitely many times.

Furthermore F. Cohen proved that L can be taken to be 2k when n = 2 [12]. This allowed him to construct a map

σ_{n} : W (n) ⟶ Ω^{2p}W (n + 1)

which can be considered to be the “secondary double-suspension”, where W (n) is the homotopy theoretic fiber of the double suspension map

E^{2} : S^{2n−1} ⟶ Ω^{2}S^{2n+1}.

σ_{n} played an essential role in the work of Mahowald and Thompson [39, 48] where they determined the unstable v_{1}-periodic homotopy groups of spheres.

B. Gray began a systematic study in this direction in [28]. He introduced the notion of EHP spectra which is an unstable way of studying Toda-Smith spectra V (n) in the fashion of Cohen-Moore-Neisendorfer [18, 19, 20]. In order his program to be accomplished, we need to find various unstable maps between loop spaces. As in the case of the classical EHP sequence, those maps could be constructed by using unstable splitting of loop spaces. In fact, Gray is very close to proving the existence of an EHP spectrum for V (0). His construction would be completed, if we could prove Ω^{2}Σ^{2}X localized at an odd prime splits after suspending twice, which is conjectured in [28]. However nothing is known about localization of Ω^{n}Σ^{n}X.

4 Constructing Maps and Spaces

As we have seen in the previous section, little cube model played an important role in the construction and applications of the stable splittings of iterated loop spaces.

It is often helpful to have extra geometric informations even though what we need is abstract homotopy-theoretic results.

As is suggested by Quillen’s work on closed model category, homotopy theory (stable or unstable) is fairly formal. It can be axiomatized nicely. But we need concrete models for practical applications.

Little cubes can be used to construct important spaces and maps. The followings are some of the examples.

4.1 Fibrations

May proved the Approximation theorem (Theorem 3.1.1) by constructing a little cube model for the path-loop fibration:

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

Definition 4.1.1 For a pair of pointed space (X, A), define E_{n}(X, A) to be the subspace of C_{n}(X) consisting of elements (c_{1}, ⋯, c_{j}; x_{1}, ⋯, x_{j}) satisfying the following properties: if x_{i_{1}}, ⋯, x_{i_{ℓ}} ∉ A then

(pr_{n−1}(c_{i_{1}}), ⋯, pr_{n−1}(c_{i_{ℓ}})) ∈ C_{n−1}(ℓ),

where pr_{n−1} is the projection onto the last (n − 1)-coordinates.

Theorem 4.1.2 (May) 1. If X is a pointed connected space with nondegenerate basepoint, we have a quasifibration

C_{n}(X) ⟶ E_{n}(CX, X) ⟶ C_{n−1}(ΣX) (2)

and natural maps

α : C_{n}(X) ⟶ Ω^{n}Σ^{n}X
α̃ : E_{n}(CX, X) ⟶ P Ω^{n−1}Σ^{n}X

making the following diagram commutative up to homotopy

[figure: C_{n}(X); E_{n}(CX, X); C_{n−1}(ΣX); Ω^{n}Σ^{n}X; P Ω^{n−1}Σ^{n}X; Ω^{n−1}Σ^{n}X.; α; α̃; α]

2. E_{n}(CX, X) is contractible. Therefore α is a weak homotopy equivalence.

Since path-loop fibration is a principal fibration with contractible total space, principal fibrations with fiber Ω^{n}Σ^{n}X are pull-back of the path-loop fibration. Thus it is natural to expect to obtain a little cube model of such a fibration by pulling back May’s quasifibration (2). Unfortunately, however, taking a pull-back in general does not preserve quasifibration.

The quasifibration (2) is not just a quasifibration. The base space is filtered and the quasifibration satisfies a kind of local-triviality on each successive difference of the filtration.

Thanks to this additional structure, a pull-back of (2) has a chance to be a quasifibration. In fact, S.-C. Wong proved the following [49]:

Definition 4.1.3 For a pointed space X, define W (k, n, ΣX) to be the homotopy fiber of the Freudenthal suspension map:

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

Namely W (k, n, ΣX) is defined by the following pull-back diagram:

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

Definition 4.1.4 For a pair of pointed spaces (X, A), define ξ(k, n; X, A) by the following pull-back diagram:

[figure: ξ(k, n; X, A); E_{n+k−1}(X, A); C_{k−1}(X/A); C_{n+k−1}(X/A).; π(k,n); π_{n+k}; σ^{n}]

Theorem 4.1.5 (Wong) For k ≥ 1, n ≥ 0 and a strong NDR pair (X, A),

π(k, n) : ξ(k, n; X, A) ⟶ C_{k−1}(X/A)

is a quasifibration with fiber C_{n+k}(A). Thus we have a weak homotopy equivalence

ξ(k, n; X, A) ≃ W (k, n; X/A).

Wong used ξ(k, n; X, A) to prove a stable splitting of W (k, n; X/A). Since the secondary-suspension σ_{n} was constructed by splitting off the p-adic piece of Ω^{2}S^{2n+1} after the 2p-fold suspension, it would be interesting if we could desuspend Wong’s stable splitting to construct “tertiary suspension map”.

Wong’s idea can be also applied to the following case.

Since C_{n}(X) has a structure of a monoid by concatenation and the equivalence

C_{n}(X) ≃ Ω^{n}Σ^{n}X

is an equivalence of Hopf spaces, the p-th power map

p× : Ω^{n}Σ^{n}X ⟶ Ω^{n}Σ^{n}X

is equivalent to the p-th power map on C_{n}(X). Thus we have a diagram

[figure: ?; E_{n}(CX, X); C_{n}(ΣX); C_{n}(ΣX).; π_{n}; p×]

in which all maps are defined in terms of little cubes.

The homotopy fiber of p× : Ω^{n}Σ^{n+1}X ⟶ Ω^{n}Σ^{n+1}X is the mapping space from the mod p Moore space Map_{∗}(P^{n+1}(p), Σ^{n+1}X). Following Wong’s idea K. Iwama recently proved the following [29].

Theorem 4.1.6 Let X be a pointed space with (X, ∗) a strong NDR pair. Define E_{n}(p)(CX, X) by the following pull-back diagram

[figure: E_{n}(p)(CX, X); E_{n}(CX, X); C_{n}(ΣX); C_{n}(ΣX).; π_{n}(p); π_{n}; p×]

Then π_{n}(p) is a quasifibration with fiber C_{n+1}(X). Therefore we have a weak homotopy equivalence

E_{n}(p)(CX, X) ≃ Map_{∗}(P^{n+1}(p), Σ^{n+1}X),

where P^{n+1}(p) = S^{n} ∪_{p} e^{n+1}.

It is not known if E_{n}(p)(CX, X) has a stable splitting. Although the proof of the above theorem is parallel to that of Wong’s, the proof of stable splitting by Wong cannot be applied in this case. The difficulty comes from the fact that the p-th power map on C_{n}(ΣX) does not preserve the filtration.

4.2 Kahn-Priddy Transfer

As the Snaith splitting suggests, little cubes are useful for handling stable maps in concrete ways. Another example of applications of little cube in stable homotopy theory is the construction of transfer by D.S. Kahn and S.B. Priddy.

Transfer is a map going to the wrong direction in the homology or cohomology, classically in group cohomology. It is a well-known fact that they can be realized as stable maps between spaces. One of the examples is constructed by Kahn and Priddy by using the little cube model of Ω^{∞}Σ^{∞}X [30, 31].

Suppose we are given an N-fold covering space

p : E ⟶ B.

The symmetric group of N-letters acts on each fiber by deck transformation and p can be considered to be a fiber bundle with fiber π_{1}(B)/π_{1}(E) and structure group Σ_{N}. Since C_{∞}(N) is a contractible space with a free Σ_{N}-action, the projection

π : C_{∞}(N) ⟶ C_{∞}(N)/Σ_{N}

is a universal Σ_{N}-bundle. Thus we have the following pull-back diagram

[figure: E; C_{∞}(N) ×_{Σ_{N}} (π_{1}(B)/π_{1}(E))^{N}; B; C_{∞}(N)/Σ_{N}.; φ; p; π; φ̄]

Define

Φ : B ⟶ C_{∞}(N) ×_{Σ_{N}} E^{N}

by

Φ(p(x)) = (φ(x), xτ_{1}, ⋯, xτ_{N})

where {τ_{1}, ⋯, τ_{N}} is a choice of coset representatives of π_{1}(B)/π_{1}(E). Composed with

C_{∞}(N) ×_{Σ_{N}} E^{N} ⟶ C_{∞}(E)

we have

B_{+} ⟶^{Φ} C_{∞}(E_{+}) ≃ Ω^{∞}Σ^{∞}(E_{+}).

The Kahn-Priddy transfer is the stable map adjoint to this map:

p^{!} : Σ^{∞}B_{+} ⟶ Σ^{∞}E_{+}.

Let p be a prime number. Kahn and Priddy applied this construction to the covering

EΣ_{p} ⟶ BΣ_{p}

to get the famous Kahn-Priddy theorem which states

π_{∗}^{S}(BΣ_{p}) ⟶ π_{∗}^{S}(S^{0})

is surjective on the p-primary components.

4.3 Strong Convergent Cobar Spectral Sequence

Little cubes can be used to construct a spectral sequence converging to the homology of iterated loop spaces:

(3) E^{2} ≅ Cotor^{h_{∗}(Ω^{n−1}Σ^{n}X)}(h_{∗}, h_{∗}) ⟹ h_{∗}(Ω^{n}Σ^{n}X).

The following is a quick review of the construction by the author [46]. Throughout this subsection h_{∗}(−) denotes a multiplicative homology theory.

In order to construct a spectral sequence (3), the most natural idea would be to try to find a filtration on Ω^{n}Σ^{n}X with which the E^{1}-term of the resulting spectral sequence becomes the algebraic cobar construction on the coalgebra h_{∗}(Ω^{n−1}Σ^{n}X). Thanks to the stable splitting (1), it enough to define a filtration on each C_{n}(j), separately.

Suppose we have a filtration {F_{−q}C_{n}(j)} on each C_{n}(j). Define

F_{−s}C_{n}(X) = ⋁_{j} F_{−s}C_{n}(j)_{+} ∧_{Σ_{j}} X^{∧j}.

{F_{−s}C_{n}(X)} is a stable filtration for Ω^{n}Σ^{n}X. The E^{1}-term of the spectral sequence defined by this filtration is

E_{−s,t}^{1} = h_{−s+t}(F_{−s}C_{n}(X), F_{−s−1}C_{n}(X))
= ⨁_{j} h_{−s+t}(F_{−s}C_{n}(j)_{+} ∧_{Σ_{j}} X^{∧j}, F_{−s−1}C_{n}(j)_{+} ∧_{Σ_{j}} X^{∧j})
= ⨁_{j} h̃_{−s+t}(F_{−s}C_{n}(j)/F_{−s−1}C_{n}(j) ∧_{Σ_{j}} X^{∧j}).

On the other hand, the tensor algebra on h̃_{∗}(Ω^{n−1}Σ^{n}X) has summands of the following form:

h̃_{∗}((C_{n−1}(j_{1}) × ⋯ × C_{n−1}(j_{s}))_{+} ∧_{Σ_{j_{1}}×···×Σ_{js}} (ΣX)^{∧(j_{1}+···+j_{s})}).

Thus what we need is filtrations on {C_{n}(j)} so that F_{−s}C_{n}(j) − F_{−s−1}C_{n}(j) becomes s vertically aligned stacks of cubes. In order to define such filtrations, we need auxiliary functions on C_{n}(j).

Let

pr_{1} : C_{n}(1) ⟶ C_{1}(1)

be the map induced by the projection onto the first coordinate. It is not difficult to find a function

d : C_{n}(1) × C_{n}(1) ⟶ [0, 1]

with the following properties.

d(c, c′) = 0 ⟺ pr_{1}(c)(\frac{1}{2}) ∉ pr_{1}(c′)([0, 1]) or pr_{1}(c′)(\frac{1}{2}) ∉ pr_{1}(c)([0, 1])
d(c, c′) = 1 ⟺ pr_{1}(c)(\frac{1}{2}) = pr_{1}(c′)(\frac{1}{2}).

For c = (c_{1}, ⋯, c_{j}) ∈ C_{n}(j), a stack in c is a subset of {c_{1}, ⋯, c_{j}}. We say that a stack {c_{i}|i ∈ S} is stable under gravity if and only if d(c_{i_{1}}, c_{i_{2}}) ≠ 0 for any i_{1}, i_{2} ∈ S.

With these terminologies we define a filtration on little cubes as follows.

Definition 4.3.1 Let F_{0}C_{n}(j) = F_{−1}C_{n}(j) = C_{n}(j). For q > 0, c = (c_{1}, ⋯, c_{j}) ∈ F_{−s−1}C_{n}(j) if and only if, for any partition {1, ⋯, j} = S_{1} ∐ ⋯ ∐ S_{q} with S_{1} ≠ ∅, ⋯, S_{q} ≠ ∅, at least one of the stacks corresponding to S_{1}, ⋯, S_{q} is not stable under gravity.

This filtration is called the gravity filtration. The spectral sequence induced by this filtration is called the gravity spectral sequence.

On the contrary to our intension, it is not known whether the E^{1}- term of the spectral sequence defined by this filtration is isomorphic to the cobar construction. However the following fact proved in [46] is enough to identify the E^{2}-term with Cotor.

Theorem 4.3.2 There exists a stable filtration on E_{n}(CX, X) with the following isomorphism of differential graded h_{∗}-modules, if h_{∗}(Ω^{n−1}Σ^{n}X) is flat over h_{∗}:

E^{1}(E_{n}) ≅ E^{1}(C_{n}) ⊗_{h_{∗}} h_{∗}(C_{n−1}(ΣX)),

where {E^{r}(E_{n})} is the spectral sequence induced by the filtration on E_{n}(CX, X) and {E^{r}(C_{n})} is the spectral sequence induced by the gravity filtration on C_{n}(X).

Furthermore (E^{1}(E_{n}), d^{1}) is acyclic.

Corollary 4.3.3 If h_{∗}(Ω^{n−1}Σ^{n}X) is flat over h_{∗}, we have the following isomorphism

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

For a pointed space X, if h_{∗}(X) is a flat h_{∗}-module, we have a spectral sequence, so-called the classical Eilenberg-Moore spectral sequence with

E^{2} ≅ Cotor^{h_{∗}(X)}(h_{∗}, h_{∗}).

The E^{∞}-term of this spectral sequence is not in general directly related to h_{∗}(ΩX). However, the gravity spectral sequence does calculate h_{∗}(Ω^{n}Σ^{n}X), since it is a direct sum of the spectral sequences defined by finite filtrations.

Theorem 4.3.4 The gravity spectral sequence converges to h_{∗}(Ω^{n}Σ^{n}X).

Although the gravity spectral sequence has this important property, it also has a disadvantage: the E^{1}-term is mysterious. In order to use the gravity spectral sequence for practical computations, it is important to find good generators in the E^{2}-term. In the case of the classical Eilenberg-Moore spectral sequence, we have the Dyer-Lashof operations which are defined in terms of the cobar construction. The following fact helps us to find generators in the E^{2}-term of the gravity spectral sequence.

Theorem 4.3.5 ([47]) If h_{∗}(Ω^{n−1}Σ^{n}X) is flat over h_{∗}, the gravity spectral sequence is isomorphic to the classical Eilenberg-Moore spectral sequence from the E^{2}-term on.

4.4 Little Cubes with Overlappings Allowed

We have been considering little cubes disjoint from each other, so far. If we remove this disjointness condition, we obtain loop spaces of different types.

Definition 4.4.1 Define D_{n}(j) = C_{n}(1)^{j}. This is the space of j little n-cubes with overlappings allowed.

c_{1}c_{2}  c_{3} c_{4}

The symmetric group of j-letters acts on D_{n}(j) by renumbering of cubes and the definition of C_{n}(X) can be applied without modification to get a functor D_{n}(X).

Since the little cube operad C_{n} acts on D_{n}(X) in a natural way, D_{n}(X) is an n-fold loop space by the recognition principle of May. In fact, we can easily see that this is equivalent to a classical construction due to Dold and Thom, i.e. infinite symmetric product, and thus it has a structure of an infinite loop space.

Definition 4.4.2 For a pointed space X and a nonnegative integer i, define

SP^{i}(X) = (∐_{j=0}^{i} X^{j}/Σ_{j}) / ∼

where the equivalence relation “∼” is generated by the relation removing basepoint.

SP^{i}(X) is called the i-th symmetric product of X. When i = ∞, it is called the infinite symmetric product.

Since there is no restriction on the movements of cubes, D_{n}(j) is Σ_{j}-equivariantly contractible. Thus we have

Proposition 4.4.3 For any pointed space X, D_{n}(X) is homotopy equivalent to SP^{∞}(X).

The following is a classical theorem of Dold and Thom [25],

Theorem 4.4.4 For a pointed space X with a nondegenerate basepoint, we have the following natural isomorphism

π_{n}(SP^{∞}(X)) ≅ H̃_{n}(X)

As an immediate corollary, we have

Theorem 4.4.5 For a pointed CW-complex X of finite type, we have the following homotopy equivalence

SP^{∞}(X) ≃ ∏_{n=0}^{∞} K(π_{n}(X), n).

C_{n}(X) and D_{n}(X) are two extreme cases: cubes are disjoint from each other in C_{n}(j), while any kinds of overlapping are allowed in D_{n}(j). π_{∗}(C_{∞}(X)) is the stable homotopy groups of X, while π_{∗}(D_{∞}(X)) is the (reduced) integral homology groups of X.

Let us consider the following intermediate objects.

Definition 4.4.6 Let D_{n}^{i}(j) be the subspace of D_{n}(j), consisting of j little cube (c_{1}, ⋯, c_{j}) satisfying the following property: each t ∈ I^{n} is contained in the interior of the image of at most i cubes.

D_{n}^{i}(j) inherits the action of Σ_{j}. For a pointed space X, define D_{n}^{i}(X) in the same way as C_{n}(X) is defined.

Now we have successive inclusions

C_{n}(X) = D_{n}^{0}(X) ⊂ D_{n}^{1}(X) ⊂ ⋯ ⊂ D_{n}^{∞}(X) = D_{n}(X).

The commutativity of the following diagram

[figure: C_{n}(X); D_{n}(X); X; ⊂]

and the fact that the induced map on homotopy groups

π_{∗}(X) ⟶ π_{∗}(D_{n}(X)) ≅ π_{∗}(SP^{∞}(X)) ≅ H̃_{∗}(X)

coincides with the Hurewicz homomorphism implies that the Hurewicz homomorphism factors through π_{∗}(D_{n}^{i}(X))

[figure: π_{∗}(D_{n}^{i}(X)); H̃_{∗}(X); π_{∗}(X).; ⊂]

F. Kato determined the homotopy type of D_{n}^{i}(X) [33].

Theorem 4.4.7 For a path-connected pointed space X with (X, ∗) a strong NDR pair, we have the following weak homotopy equivalence

D_{n}^{i}(X) ≃ Ω^{n}SP^{i}Σ^{n}X.

The idea of the proof is to extend May’s construction of the quasifibration (2).

Since

D_{0}^{i}(j) = \{ \begin{matrix}∗ & if j ≤ i \\ ∅ & if j > i \end{matrix}

we have

D_{0}^{i}(X) = SP^{i}(X).

Once we have a quasifibration

D_{n}^{i}(X) ⟶ E ⟶ D_{n−1}^{i}(ΣX)

with E contractible, we have the desired weak homotopy equivalence

D_{n}^{i}(X) ≃ ΩD_{n−1}^{i}(ΣX) ≃ ⋯ ≃ Ω^{n}D_{0}^{i}(Σ^{n}X) = Ω^{n}SP^{i}Σ^{n}X.

Definition 4.4.8 For a pair of pointed spaces (X, A), define E_{n}^{i}(X, A) to be the subspace of C_{n}^{i}(X) consisting of elements (c_{1}, ⋯, c_{j}; x_{1}, ⋯, x_{j}) satisfying the following properties: if x_{i_{1}}, ⋯, x_{i_{ℓ}} ∉ A then

(pr_{n−1}(c_{i_{1}}), ⋯, pr_{n−1}(c_{i_{ℓ}})) ∈ C_{n−1}^{i}(ℓ).

Proposition 4.4.9 For a pointed space X with (X, ∗) a strong NDR pair, the projection

E_{n}^{i}(CX, X) ⟶ D_{n−1}^{i}(ΣX)

is a quasifibration with fiber D_{n}^{i}(X).

Furthermore E_{n}^{i}(CX, X) is contractible.

Sadok Kallel [32] independently studied an analogous construction C^{i}(R^{n}; X) by using (labelled) configuration space of points in R^{n}, instead of little cubes, and proved a homotopy equivalence

C^{i}(R^{n}; X) ≃ Ω^{n}SP^{i}Σ^{n}X

for connected CW complexes together with a “delooped version” of this homotopy equivalence.

We should point out that most of the constructions using little ncubes can be also done by using configuration space of points in R^{n}. One of the advantages of using configuration space of points rather than little cubes is that R^{n} can be replaced with any space M. Kallel also studied C^{d}(M; X) where M is a stably parallelizable smooth manifold with nonempty boundary.

5 Problems

We conclude this article with open problems related to little cubes.

1. Find a little cube model of the fiber of the secondary suspension

σ_{n} : W (n) ⟶ Ω^{2p}W (n + 1).

2. Find a stable splitting of Map_{∗}(P^{n}(p), Σ^{n}X).

3. Prove that, localized away from 2, Ω^{2}Σ^{2}X splits into a wedge of the (localized) Snaith summands after suspending twice.

4. For each prime p, find a combinatorial model for Ω^{n}Σ^{n}X localized at or away from p.

5. Compute the homology of Ω^{n}SP^{i}Σ^{n}X.

24 DAI TAMAKI

Acknowledgment

The author is very grateful to Jesus Gonzalez, CINVESTAV, for inviting him to write this article for Morfismos, which gave him a good chance to review the development of the theory of little cubes. He would also like to thank the referees for informing him of Kallel’s work and for pointing out an ambiguous assumption in a theorem.

Dai Tamaki

Department of Mathematical Sciences Faculty of Science, Shinshu University 3-1-1 Asahi, Matsumoto, 390-8621 JAPAN rivulus@math.shinshu-u.ac.jp

References

[1] Kudo T. and Araki S., Topology of H_{n}-spaces and H-squaring operations., Memoirs of the Faculty of Science, Kyusyu Univ., Ser. A, 10 (1956), 85–120.

[2] Barratt M.G. and Eccles P.J., Γ^{+}-structures – I: A free group functor for stable homotopy theory., Topology, 13 (1974), 25–45.

[3] Barratt M.G. and Eccles P.J., Γ^{+}-structures – II: A recognition principle for infinite loop spaces., Topology, 13 (1974), 113–126.

[4] Barratt M.G. and Eccles P.J., Γ^{+}-structures – III: The stable structure of Ω^{∞}Σ^{∞}A., Topology, 13 (1974), 199–207.

[5] Boardman J.M. and Vogt R.M., Homotopy-everything H-spaces., Bulletin of the American Mathematical Society, 74 (1968), 1117–1122.

[6] Boardman J.M. and Vogt R.M., Homotopy Invariant Algebraic Structures on Topological Spaces., Lecture Notes in Mathematics, vol. 347, Springer-Verlag, 1973.

[7] Brown E.H. and Gitler S., A spectrum whose cohomology is a certain cyclic module over the Steenrod algbebra., Topology, 12 (1973), 283–295.

[8] Brown E.H. and Peterson F.P., On the stable decomposition of Ω^{2}S^{r+2}., Transactions of the American Mathematical Society, 243 (1978), 287–298.

[9] Caruso Jeffrey L., A simpler approximation to QX., Transactions of the American Mathematical Society, 265 (1981), 163–167.

[10] Caruso J., Cohen F.R., May J.P. and Taylor L.R., James maps, Segal maps, and the Kahn-Priddy theorem., Transactions of the American Mathematical Society, 281 (1984), 243–283.

[11] Caruso J. and Waner S., An approximation to Ω^{n}Σ^{n}X., Transactions of the American Mathematical Society, 265 (1981), 147–162.

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

LITTLE CUBES AND HOMOTOPY THEORY 25

[13] Cohen F.R., Cohen R.L., Kuhn N.J. and Neisendorfer J.L., Bundles over configuration spaces., Pacific Journal of Mathematics, 104 (1983), 47–54.

[14] Cohen F.R., Mahowald M.E. and Milgram M.J., The stable decomposition for the double loop space of a sphere., Proceedings of Symposia in Pure Mathematics vol. 32 (1978), 225–228.

[15] Cohen F.R., May J.P. and Taylor L.R., Splitting of certain spaces CX., Mathematical Proceedings of the Cambridge Philosophical Society, 84 (1978), 465– 496.

[16] Cohen F.R., May J.P. and Taylor L.R., Splitting of some more spaces., Mathematical Proceedings of the Cambridge Philosophical Society, 86 (1979), 227– 236.

[17] Cohen F.R., May J.P. and Taylor L.R., James maps and E_{n} rings spaces., Transactions of the American Mathematical Society, 281 (1984), 285–295.

[18] Cohen F.R., Moore, J.C. and Neisendorfer J.L., Torsion in homotopy groups., Annals of Mathematics, 109 (1979), 121–168.

[19] Cohen F.R., Moore, J.C. and Neisendorfer J.L., The double suspension and exponents of the homotopy groups of spheres., Annals of Mathematics, 110 (1979), 549–565.

[20] Cohen F.R., Moore, J.C. and Neisendorfer J.L., Exponents in homotopy theory., In Algebraic Topology and Algebraic K-theory, Annals of Mathematical Studies No. 113, Princeton University Press, 1987, 3–34.

[21] Cohen F.R., Lada T.J. and May J.P., The Homology of Iterated Loop Spaces., Lecture Notes in Mathematics, vol. 533, Springer-Verlag, 1976.

[22] Cohen R.L. The geometry of Ω^{2}S^{3} and braid orientation., Inventiones Mathematicae, 54 (1979), 53–67.

[23] Cohen R.L. Odd primary infinite families in stable homotopy theory., Memoirs of the American Mathematical Society, No. 242, 1981.

[24] Cohen R.L. Stable proofs of stable splittings., Mathematical Proceedings of the Cambridge Philosophical Society, 88 (1980), 149–151.

[25] Dold A. and Thom R., Quasifaserungen und unendliche symmetrische Produkte., Annals of Mathematics, 67 (1958), 239–281.

[26] Dyer E. and Lashof R.K., Homology of iterated loop spaces., American Journal of Mathematics, 84 (1962), 35–88.

[27] James I., Reduced product spaces., Annals of Mathematics, 62 (1955), 170–197.

[28] Gray B., EHP spectra and periodicity. I: geometric constructions., Transactions of the American Mathematical Society, 340 (1993), 595–616

[29] Iwama K., Constructing mapping spaces from Moore spaces via little cubes., Master’s thesis, Department of Mathematics, Shinshu University, March 1998.

[30] Kahn D.S. and Priddy S.B., Applications of the transfer to stable homotopy theory., Bulletin of the American Mathematical Society, 78 (1972), 981–987.

[31] Kahn D.S. and Priddy S.B., The transfer and stable homotopy theory., Mathematical Proceedings of the Cambridge Philosophical Society, 83 (1978), 103–

26 DAI TAMAKI

[32] Kallel S., An interpolation between homology and stable homotopy., preprint.

[33] Kato F., On the space realizing the intermediate stages of the Hurewicz homomorphism., Master’s thesis, Department of Mathematics, Shinshu University, March 1996.

[34] Kuhn Nicholas J., The geometry of the James-Hopf maps., Pacific Journal of Mathematics, 102 (1982), 397–412.

[35] Kuhn Nicholas J., The homology of the James-Hopf maps., Illinois Journal of Mathematics, 27 (1983), 315–333.

[36] Loday J.-L., Stasheff J.D. and Voronov A.A. ed. Operads: Proceedings of Renaissance Conferences., Contemporary Mathematics 202, American Mathematical Society, 1997.

[37] Lewis L.G., May J.P. and Steinberger M. Equivariant Stable Homotopy Theory., Lecture Notes in Mathematcis, vol. 1213, Springer-Verlag, 1986.

[38] Mahowald M.E., A new infinite family in _{2}π_{∗}^{S}., Topology, 16 (1977), 249–256.

[39] Mahowald M.E., The image of J and the EHP sequence., Annals of Mathematics, 116 (1982), 65–112.

[40] May J.P., The Geometry of Iterated Loop Spaces., Lecture Notes in Mathematics, vol. 271, Springer-Verlag, 1972.

[41] Milgram R.J., Iterated loop spaces., Annals of Mathematics, 84 (1966), 386– 403.

[42] Milnor J.W., On the construction F K., Reprinted in J.F. Adams, “Algebraic Topology – A Student’s Guide”, London Mahtematical Society Lecture Notes Series 4, Camridge University Press, 1972.

[43] Segal G.B., Configuration spaces and iterated loop spaces., Inventiones Mathematicae, 21 (1973), 213–221.

[44] Smith J.H., Simplicial group models for Ω^{n}S^{n}X., Israel Journal of Mathematics, 66 (1989), 330–350.

[45] Snaith V.P.A., Stable decomposition for Ω^{n}S^{n}X., Journal of the London Mathematical Society (2), 7 (1974), 577–583.

[46] Tamaki D. A dual Rothenberg-Steenrod spectral sequence., Topology, 33 (1994), 631–662.

[47] Tamaki D. Remarks on the cobar-type Eilenberg-Moore spectral sequences., preprint.

[48] Thompson R.D., The v_{1}-periodic homotopy groups of an unstable sphere at odd primes., Transactions of the American Mathematical Society, 319 (1990), 535–559.

[49] Wong S.-C., The fiber of the iterated Freudenthal suspension., Mathematische Zeitschrift, 215 (1994), 377–414.