Remarks on the Cobar-type Eilenberg-Moore Spectral Sequences

Dai Tamaki Department of Mathematics Faculty of Science Shinshu University Matsumoto, Nagano Pref. 390 Japan

Abstract

In this note, we compare various constructions of cobar-type spectral sequences.We prove that the constructions by Rector, L.Smith, and the author give the same spectral sequence from the E^{2}- term on. Since Rector proved his spectral sequence is isomorphic to the classical Eilenberg-Moore spectral sequence, all the known cobar-type spectral sequences are isomorphic.

Introduction

For a pointed space X, homology rings of the iterated loop spaces on X, Ω^{n}X for n = 1, 2, …, provide us with good information on the homotopy groups of X.

There are several systematic tools for computing the homology of the loop space ΩX from the homology of X. One of the most efficient is the cobar-type Eilenberg-Moore spectral sequence [6, 7]. For reasonably good homology theory h_{∗}(−), it is usually written in the following form.

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

One of the difficulties in using the cobar-type Eilenberg-Moore spectral sequence is the convergence problem. In general, the E^{∞}-term of the spectral sequence is not directly related to h_{∗}(ΩX). For singular homology, W. Dwyer [5] obtained a nice description of the E^{∞}-term. For connective homology theory, A. Bousfield [2] reduced the convergence problem to that of singular homology. As is suggested by the computation of the Morava K-theory of the Eilenberg-Mac Lane spaces [1, 10], it is not easy to discuss the convergence problem for nonconnective homology theory.

However, for loop spaces of the form Ω^{n}Σ^{n}X, the author [14] constructed a strongly convergent spectral sequence, called the gravity spectral sequence, for any homology theory, which is of the following form,

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

if h_{∗}(−) is multiplicative and h_{∗}(Ω^{n−1}Σ^{n}X) is h_{∗}-flat. One of the disadvantages of this spectral sequence is that the E^{1}-term is not necessarily the cobar construction on h̃_{∗}(Ω^{n−1}Σ^{n}X), although the E^{2}-term is Cotor. For practical computations, it is important to have canonical generators of the E^{2}-term given by elements in the cobar construction.

The best way of resolving these difficulties in the classical Eilenberg-Moore spectral sequence and the gravity spectral sequence is to prove that the gravity spectral sequence is isomorophic to the classical Eilenberg-Moore spectral sequence from the E^{2}-term on. There are other spectral sequences with E^{2}- term Cotor^{h_{∗}(X)}(h_{∗}, h_{∗}) and abutting to h_{∗}(ΩX).

The purpose of this paper is to show that gravity spectral sequence (1), together with other known constructions of cobar-type spectral sequences, is isomorphic to the classical Eilenberg-Moore spectral sequencefrom the E^{2}-term on.

The paper is organized as follows.

In §1, we briefly recall the original construction of the Eilenberg-Moore spectral sequence by Eilenberg and Moore.

In §2, Rector’s geometric cobar construction is reviewed.

In §3, Another way of obtaining a cobar spectral sequence out of the geometric cobar construction, due to Dwyer, is reviewed.

In §4, basic constructions in the category (Spaces /B)_{∗} and their properties are recalled from [13, 12]. In §5, Smith’s construction of Eilenberg-Moore spectral sequence in terms of display in (Spaces /B)_{∗}

is explained with some simplifications.

In §6, we prove that Rector’s construction comes from a display and hence it is equivalent to Smith’s construction.

In §7, we prove that the cobar-type spectral sequence in [14], in fact, is induced from a display. Thus it is proved that all the known cobar-type Eilenberg-Moore spectral sequence are isomorphic from the E^{2}-term on.

1 The Original Construction of the Eilenberg-Moore Spectral Sequence

Originally Eilenberg and Moore studied chain complexes (or more generally, differential graded objects in an abelian category) instead of spaces [6, 7]. Fix a ground ring k. For a differential graded coalgebra C over k and left and right differential graded comodules M and N over C, they defined a differential graded k-module T (M, C, N) as the total complex of the algebraic cobar construction.

Consider a pull-back diagram of spaces.

[figure: F; X; Y; B; p; f]  (2)

Let S_{∗}(−; k) be the singular chain complex functor with coefficients in k. By the Eilenberg-Zilber theorem, S_{∗}(B; k) has a structure of differential graded coalgebra over k. The maps f and g induces maps of differential graded coalgebras

f_{∗} : S_{∗}(Y; k) ⟶ S_{∗}(B; k)
p_{∗} : S_{∗}(X; k) ⟶ S_{∗}(B; k).

Composed with diagonals, these maps induce left and right differential graded comodule structures over S_{∗}(B; k) on S_{∗}(Y; k) and S_{∗}(X; k), respectively.

S_{∗}(Y; k) ⟶^{Δ_{∗}} S_{∗}(Y; k) ⊗ S_{∗}(Y; k) ⟶^{f_{∗}⊗id} S_{∗}(B; k) ⊗ S_{∗}(Y; k)
S_{∗}(X; k) ⟶^{Δ_{∗}} S_{∗}(X; k) ⊗ S_{∗}(X; k) ⟶^{id⊗g_{∗}} S_{∗}(X; k) ⊗ S_{∗}(B; k).

Eilenberg and Moore proved

Theorem 1.1. For a pull-back diagram (2), if p : X ⟶ B is a fibration, there is a natural isomorphism

H_{∗}(F; k) ≃ H_{∗}(T (S_{∗}(Y; k), S_{∗}(B; k), S_{∗}(X; k)))  (3)

The algebraic cobar construction naturally induces a filtration on

T (S_{∗}(Y; k), S_{∗}(B; k), S_{∗}(X; k))

The original Eilenberg-Moore spectral sequence is the spectral sequence induced from this filtration by taking homology.

2 Rector’s Geometric Cobar Construction

Since the original Eilenberg-Moore spectral sequence is defined by a filtration on a chain complex (differential graded module), it only works for singular homology theory. Furthermore, it is not easy to show the natural action of the Steenrod operations on the E^{2}-term commutes with differentials and induces the action on the E^{∞}-term which agrees with the one on H_{∗}(F).

One of the ways to avoid these difficulties is to find a space-level construction of the spectral sequence. Since the singular chain functor S_{∗}(−) induces an equivalence between the homotopy category of CW - complexes and the homotopy category of simplicial abelian groups, there should be a “geometric cobar construction” which gives a geometric version of the equivalence (3). This is done by Rector [11] by using the cosimplicial technology developed by Bousfield and Kan [3].

Rector’s geometric cobar construction is exactly a dual to the simplical bar construction of the classifying space of an associative topological monoid [9].

For a pointed space B, let

Δ : B ⟶ B × B

be the diagonal map. This is cocommutative and coassociative and has a counit. We can then form a cosimplicial space Ω^{∗}(Y, B, X) for a pull-back diagram (2)

Ω^{0}(Y, B, X) ⟶^{δ^{0}} ; ⟶^{δ^{1}} ; ⟵^{σ^{1}} Ω^{1}(Y, B, X) ⟶^{δ^{0}} ; ⟶^{δ^{1}} ; ⟶^{δ^{2}} ; ⟵^{σ^{1}} ; ⟵^{σ^{2}} Ω^{2}(Y, B, X) ⋯

where

Ω^{n}(Y, B, X) = Y × B^{n} × X

and

δ^{i}(y, b_{1}, ⋯, b_{n}, x) = \{ \begin{matrix}(y, f(y), b_{1}, ⋯, b_{n}, x) & if i = 0 \\ (y, b_{1}, ⋯, b_{i}, b_{i}, ⋯, b_{n}, x) & if 1 ≤ i ≤ n \\ (y, b_{1}, ⋯, b_{n}, p(x), x) & if i = n \end{matrix}
σ^{i}(y, b_{1}, ⋯, b_{n}, x) = (y, b_{1}, ⋯, b_{i−1}, b_{i+1}, ⋯, b_{n}, x).

The geometric analogue of cohomotopy is the “Tot” functor of Bousfield and Kan.

Definition 2.1. Let Δ^{∗} be the cosimplicial space of the standard simplices. For any cosimplicial space X^{∗}, define

Tot(X^{∗}) = Map_{csSpaces}(Δ^{∗}, X^{∗}),

where csSpaces is the category of cosimplicial spaces. Tot is a covariant functor

Tot : csSpaces ⟶ Spaces

Proposition 2.2. For the pull-back diagram (2), we have a natural weak homotopy equivalence

F ≃_{w} Tot(Ω^{∗}(Y, B, X)).

Thus we obtain a geometric analogue of the Eilenberg-Moore equivalence (3). The next step is to define a filtration on Tot(Ω^{∗}(Y, B, X)), correspnding to the Eilenberg-Moore filtration on

Cotor^{S_{∗}(B)}(S_{∗}(Y), S_{∗}(X)).

Unfortunately, this is not easy. This difficulty is closely related to the convergence problem of the Eilenberg-Moore spectral sequence. Instead of filtering Tot(Ω^{∗}(Y, B, X)), Rector defined a sequence of pointed cofibrations

Ω_{0} ⟶^{φ_{−1}}  \overline{Ω}_{−1} ⟶ Ω_{−1}
Ω_{−1} ⟶^{φ_{−2}}  \overline{Ω}_{−2} ⟶ Ω_{−2}
⋮  (4)
Ω_{−n+1} ⟶^{φ_{−n}}  \overline{Ω}_{−n} ⟶ Ω_{−n}

⋮ directly from the cosimplicial cobar construction Ω^{∗}(Y, B, X) as follows.

First define

\overline{Ω}_{−n} = Ω^{n}(Y, B, X)/ Im δ^{1} ∪ ⋯ ∪ Im δ^{n}.

δ^{0} induces a well-defined map

ψ_{−n} : \overline{Ω}_{−n+1} ⟶ \overline{Ω}_{−n}.

It is easy to check that ψ_{−n−1}ψ_{−n} = ∗.

Ω_{−n}’s are defined inductively on n. Define

Ω_{0} = Ω^{0}(Y, B, X) = Y × X.

Note that \overline{Ω}_{0} = Ω_{0}/ϕ. Let φ_{−1} be the composition

Ω_{0} ⟶ Ω_{0}/ϕ = \overline{Ω}_{0} ⟶^{ψ_{−1}} \overline{Ω}_{−1}.

Let Ω_{−1} be the (reduced) mapping cone of φ_{−1}. Since ψ_{−2}ψ_{−1} = ∗, ψ_{−2} induces a well-defined map

φ_{−2} : Ω_{−1} ⟶ \overline{Ω}_{−2}.

More explicitly φ_{−2} is given by

φ_{−2}(x) = \{ \begin{matrix}ψ_{−2}(x) & if x ∈ \overline{Ω}_{−1} \\ ∗ & otherwise. \end{matrix}

From this description, it is easy to show ψ_{−3}φ_{−2} = ∗.

Inductively we obtain a map

φ_{−n} : Ω_{−n+1} ⟶ \overline{Ω}_{−n}

with ψ_{−n−1}φ_{−n} = ∗. Define Ω_{−n} to be the mapping cone of φ_{−n}.

For any homology theory h_{∗}(−), the sequence of cofibrations (4) induces an exact couple

D_{−p,q}^{1} = h̃_{q}(Ω_{−p})
E_{−p,q}^{1} = h̃_{q}(\overline{Ω}_{−p})

Rector proved, in [11],

Theorem 2.3. When h_{∗}(−) is singular homology theory, the spectral sequence associated with the above exact couple is naturally isomorphic to the original Eilenberg-Moore spectral sequence in §1.

3 Dwyer’s Modification of Rector’s Construction

With Rector’s construction, it is not easy to identify the E^{∞}-term, since the homology of each Ω_{−n} is mysterious. Dwyer modified Rector’s construction to give a nice description of the E^{∞}-term [5].

The point of departure is not to try to construct cofibrations out of a cosimplicial space. We should use a tower of fibrations instead of a sequence of cofibrations. To see why this is the case, let us consider the construction of the bar-type Eilenberg-Moore spectral sequence for the classifying space of an associative H-space G.

The multiplication on G yields a simplicial space B_{∗}(G) with B_{n}(G) = G^{n}. The classifying space of G is the geometric realization of this simplicial space

BG = |B_{∗}(G)| = (∐_{n} Δ^{n} × B_{n}(G)) / ∼

where ∼ is the well-known relation. In other words, the geometric realization of the simplicial space B_{∗}(G) is a kind of “balanced product” or “tensor product” of the cosimplicial space Δ^{∗} and the simplicial space B_{∗}(G). The skeletal filtration on Δ^{∗} induces a filtration

∗ ⊂ (BG)^{(1)} ⊂ (BG)^{(2)} ⊂ ⋯ ⊂ BG

on BG in which each successive inclusion is a cofibration if the base point of G is nondegenerate. The bar spectral sequence is obtained from this filtration by applying homology.

Thus in the case of the bar construction, the skeletal filtration on Δ^{∗} very naturally induces a sequence of cofibrations. That’s why the convergence problem of the bar spectral sequence can be solved by the standard argument.

On the other hand, for a pull-back diagram (2), we have a weak homotopy equivalence

F ≃_{w} Tot(Ω^{∗}(Y, B, X)) = Map_{csSpaces}(Δ^{∗}, Ω^{∗}(Y, B, X)).

If we apply the functor Map_{csSpaces}(−, Ω^{∗}(Y, B, X)) to the skeletal filtration on Δ^{∗}, we obtain a tower of fibrations, since Map_{csSpaces}(−, −) is contravariant in the first entry. Thus, in the case of cosimplicial spaces, the skeletal filtration on Δ^{∗} naturally induces a tower of fibrations. This is not a good news because we want to apply homology to get long exact sequences. We have to use homotopy groups in order to get long exact sequences out of a tower of fibrations.

Now it is well-known that any homology theory can be defined as homotopy groups. The oldest result in this direction is a theorem by Dold and Thom [4].

Theorem 3.1. For a pointed space X with a nondegenerate base point, there is a natural isomorphism

H̃_{∗}(X; Z) ≅ π_{∗}(SP ∞(X))

where SP ∞(X) is the infinite symmetric product on X.

More generally, if R is a finite field or a subring of Q, Bousfield and Kan [3] constructed a functor R_{∞}(−) with

H̃_{∗}(X; R) ≅ π_{∗}(R_{∞}(X)).

Note that R_{∞}(Ω^{∗}(Y, B, X)) is again a cosimplicial space because of the functoriality of R_{∞}. Thus by applying the functor Map_{csSpaces}(−, R_{∞}(Ω^{∗}(Y, B, X))) to the skeletal filtration on Δ^{∗}, we obtain a tower of fibrations. Taking homotopy groups gives us an exact couple, hence a spectral sequence. The convergence problem of the homotopy spectral sequence for a tower of fibrations is parallel to that of the homology spectral sequence for a filtered space. In this case, it is not difficult to prove that the spectral sequence converges to

π_{∗}(Tot(R_{∞}(Ω^{∗}(Y, B, X)))).

It is not clear at all, but Dwyer proved, that this group is pro-isomorphic (i.e. isomorphic as inverse systems) to H̃_{∗}(F; R) (considered as the constant inverse system), provided the pull-back diagram (2) satisfies a certain condition. This is Dwyer’s construction of cobar-type Eilenberg-Moore spectral sequence.

For a generalized homology h_{∗}(−) defined by a connective spectrum E, we have, by definition,

h̃_{∗}(X) ≅ π_{∗}(Ω^{∞}(E ∧ X)),

where Ω^{∞}(E ∧ X) is the 0-th space of the Ω-spectrum associated with E ∧ X. Hence we can use Ω^{∞}(E∧(−)) in Dwyer’s construction, instead of R_{∞}(−), to obtain the Eilenberg-Moore spectral sequence for h_{∗}(−), although we do not know how to describe the E^{∞}-term.

If h_{∗}(−) is represented by a nonconnective spectrum E,

h̃_{∗}(X) ≅ colim_{n} π_{∗+n}(Ω^{∞}(Σ^{n}E ∧ X)).

In this case we obtain a spectral sequence {E^{r}(n)} for each n by using Ω^{∞}(Σ^{n}E ∧ (−)). And then by taking the colimit of {E^{r}(n)} we obtain a spectral sequence. This is called the classical Eilenberg-Moore spectral sequence for h_{∗}(−). This is justified by the following fact.

Theorem 3.2. Dwyer’s and Rector’s constructions give the same spectral seqeuence from the E^{1}-term on.

In the case of singular homology, Dwyer explicitly described the E^{∞}-term. Bousfield studied the case of connective homology. Obviously, from the construction, it is very difficult to discuss the convergence problem in the case of nonconnective homology.

4 The Category of Pointed Spaces over B

In [12, 13], Larry Smith introduced another construction of the Eilenberg-Moore spectral sequence. His approach is closer to Rector’s than to Dwyer’s. In fact, Smith’s construction can be regarded as a generalization of Rector’s, although they seem to have discovered their constructions independently from each other.

The first step in Smith’s approach is to consider a fibration X ⟶^{p} B as an object in the category of spaces over B.

Definition 4.1. Let B be an arbitrary space. A space over B is a continuous map f : X ⟶ B. A pointed space over B is a pair of maps

f : X ⟶ B
s : B ⟶ X

with f ∘ s = id_{B}. For simplicity, we denote this object by (f, s). Morphisms between these objects are obviously defined.

If f : X ⟶ B is a space over B, we sometimes denote the “total space” X by T (f). Similarly, for (f, s) a pointed space over B, we use the notation T (f) = X if f : X ⟶ B.

The category of spaces over B is denoted by Spaces /B. The category of pointed spaces over B is denoted by (Spaces /B)_{∗}.

When B = ∗, we simply denote Spaces and Spaces_{∗} for Spaces /∗ and (Spaces /∗)_{∗}, respectively. They are the usual categories of spaces and pointed spaces, respectively.

Almost any important constructions and notions in Spaces or in Spaces_{∗} have analogies in Spaces /B or in (Spaces /B)_{∗}. To be self-contained, we record those which are used in the following sections.

Definition 4.2. For morphisms φ_{0}, φ_{1} : (f, s) ⟶ (g, t), a homotopy from φ_{0} to φ_{1} is a map

φ : T (f) × I ⟶ T (g)

which fits into the following commutative diagrams.

[figure: T (f) × I; T (g); T (f) × I; T (g); T (f); T (f) × {r}; B; B; B; B; φ; g; φ; f; =; s; =; t] for any r ∈ [0, 1]

[figure: T (f) × {0}; T (f) × I; T (f) × {1}; T (g); T (g); T (g); φ_{0}; φ; φ_{1}; =; =]

Definition 4.3. A morphism φ : (f, s) ⟶ (g, t) in (Spaces /B)_{∗} is called a cofibration in (Spaces /B)_{∗} if it has the homotopy extension property with respect to the homotopy defined above.

Definition 4.4. For a morphism φ : (f, s) ⟶ (g, t) in (Spaces /B)_{∗}, the (reduced) mapping cone of φ, denoted by (C(φ), S_{C}(φ)), is defined as follows.

T (C(φ)) = T (f) × I ⨿ T (g) / \begin{matrix}(x, 0) ∼ φ(x) & for x ∈ T (f) \\ (s(b), r) ∼ t(b) & for b ∈ B, r ∈ I \\ (x, 1) ∼ (x′, 1) & if f(x) = f(x′) for x, x′ ∈ T (f) \end{matrix}

The projection C(φ) : T (C(φ)) ⟶ B is given by f on T (f) × I and by g on T (g). The section S_{C}(φ) : B ⟶ T (C(φ)) is defined either by s or t which agree in T (C(φ)).

Definition 4.5. For an object (f, s) ∈ (Spaces /B)_{∗} the (reduced) suspension of (f, s) ∈ (Spaces /B)_{∗}, denoted by S(f, s) = (S(f), S(s)), is defined by

T (S(f)) = T (f) × I / \begin{matrix}(x, 0) ∼ (x′, 0) & if f(x) = f(x′) for x, x′ ∈ T (f) \\ (x, 1) ∼ (x′, 1) & if f(x) = f(x′) for x, x′ ∈ T (f) \\ (s(b), r) ∼ (s(b), r′) & for b ∈ B and r, r′ ∈ I \end{matrix}

and

S(f)(x, t) = f(x)
S(s)(b) = (s(b), 0)

Definition 4.6. For objects (f, s) and (g, t) in (Spaces /B)_{∗}, the smash product of (f, s) and (g, t), denoted by (f, s) ∧ (g, t) = (f ∧_{B} g, s ∧_{B} t), is defined by

T (f ∧_{B} g) = T (f) ×_{B} T (g)/(x, t(b)) ∼ (s(b), y)if f(x) = g(y) = b for x ∈ T (f), y ∈ T (g), b ∈ B

and

(f ∧_{B} g)(x, y) = f(x) = g(y)
(s ∧_{B} t)(b) = (s(b), t(b))

Lemma 4.7. If (f, s) ⟶^{φ} (g, t) is a cofibration in (Spaces /B)_{∗} is a cofibration, then so is

(h, u) ∧ (f, s) ⟶^{id∧φ} (h, u) ∧ (g, t)

with cofiber (h, u) ∧ (C(φ), S_{C}(φ)).

We fix notations for forgetful functors and their adjoints among the above categories.

Definition 4.8. Let

F : (Spaces /B)_{∗} ⟶ Spaces /B

be the functor given by forgetting sections. Its adjoint is denoted by

G : Spaces /B ⟶ (Spaces /B)_{∗}

which is given, on total spaces, by

T (G(f)) = T (f) ⨿ B.

The section is defined to be the identity map into the second component B.

Let

Γ : Spaces_{∗} ⟶ (Spaces /B)_{∗}

be the functor defined, on the total spaces, by

T (Γ(X)) = X × B.

The section is defined by the composition

B = {∗} × B ↪ X × B

where ∗ is the base point of X. Its adjoint is the functor

Φ : (Spaces /B)_{∗} ⟶ Spaces_{∗}

defined, for an object (f, s), by

Φ(f, s) = T (f)/s(B).

Definition 4.9. For any nonnegative integer n, we denote (S_{B}^{n}, s_{B}^{n}) = Γ(S^{n}).

With these functors, we can describe the definition of homotopy in (Spaces /B)_{∗} more compactly. Lemma 4.10. For morphisms φ_{0}, φ_{1} : (f, s) ⟶ (g, t), a homotopy from φ_{0} to φ_{1} is a morphism

φ : (f, s) ∧_{B} Γ(I_{+}) ⟶ (g, t)

with the following commutative diagram

[figure: (f, s) ∧_{B} Γ({0}_{+}); (f, s) ∧_{B} Γ(I_{+}); (f, s) ∧_{B} Γ({1}_{+}); (f, s); (g, t); (f, s); ∥; φ; ∥; φ_{0}; φ_{1}]

The functor Φ has a very good property. Lemma 4.11. For any cofibration

(f_{0}, s_{0}) ⟶ (f_{1}, s_{1}) ⟶ (f_{2}, s_{2})

in (Spaces /B)_{∗},

Φ(f_{0}, s_{0}) ⟶ Φ(f_{1}, s_{1}) ⟶ Φ(f_{2}, s_{2})

is a cofibration in Spaces_{∗}.

Thanks to this lemma, any (reduced) homology theory on Spaces_{∗} naturally extends to (Spaces /B)_{∗}. Definition 4.12. For any homology theory h̃_{∗}(−) on Spaces_{∗} and an object (f, s) : X ⟶ B in (Spaces /B)_{∗}, define

h_{∗}^{B}(f, s) = h̃_{∗} ∘ Φ(f, s) = h̃_{∗}(X/s(B)).

Thus we have a covariant functor

h_{∗}^{B}(−) : (Spaces /B)_{∗} ⟶ Graded Abelian Groups .

The functor h_{∗}^{B}(−) is referred to as the homology theory on (Spaces /B)_{∗} associated with h̃_{∗}(−).

Corollary 4.13. For any cofibration in (Spaces /B)_{∗}

(f_{0}, s_{0}) ⟶ (f_{1}, s_{1}) ⟶ (f_{2}, s_{2})

and a homology theory h̃_{∗}(−) on Spaces_{∗}, we have a long exact sequence.

⋯ ⟶ h_{∗}^{B}(f_{0}, s_{0}) ⟶ h_{∗}^{B}(f_{1}, s_{1}) ⟶ h_{∗}^{B}(f_{2}, s_{2}) ⟶^{∂} h_{∗−1}^{B}(f_{0}, s_{0}) ⟶ ⋯

5 Displays in (Spaces /B)_{∗} and the Eilenberg-Moore Spectral Sequences

In this section we recall the construction of a cobar-type Eilenberg-Moore spectral sequence by Larry Smith. His idea is to construct a spectral sequence out of a “filtration” in the category (Spaces /B)_{∗}, i.e. display. Smith made a lot of assumptions on (co)homology theory and the space B in his paper [13]. Most of his assumptions are for the existence of a display and the convergence of the spectral sequence. Since our purpose is to show Rector’s construction and the gravity filtration on Ω^{n}Σ^{n}X give rise to displays, we do not need these assumptions. What we really need is the following.

Assumption 5.1. Throughout the rest of this paper, h̃_{∗}(−) denotes a (reduced) multiplicative homology theory. We also assume that the external product

h̃_{∗}(B_{+}) ⊗_{h_{∗}} h̃_{∗}(X) ⟶ h̃_{∗}(B_{+} ∧ X)

is an isomorphism for any pointed space X. This condition is satisfied, for example, if h̃_{∗}(B_{+}) is h_{∗}-flat.

This condition is necessary for the following definition.

Definition 5.2. For any pointed space (f, s) over B, define a structure of left h_{∗}^{B}(S_{B}^{0}, s_{B}^{0})-comodule on h_{∗}^{B}(f, s) by the composition

h_{∗}^{B}(f, s) ≅ h̃_{∗}(Φ(f, s)) ⟶^{Δ_{∗}} h̃_{∗}(Φ(f, s) ∧ Φ(f, s)) ⟶^{Φ(∗)∧id} h̃_{∗}(Φ(S_{B}^{0}, s_{B}^{0}) ∧ Φ(f, s))
=  h̃_{∗}(B_{+} ∧ Φ(f, s))
≅  h̃_{∗}(B_{+}) ⊗_{h_{∗}} h̃_{∗}(Φ(f, s))
=  h_{∗}^{B}(S_{B}^{0}, s_{B}^{0}) ⊗_{h_{∗}} h_{∗}^{B}(f, s).

Similarly h_{∗}^{B}(f, s) also has a structure of right h_{∗}^{B}(S_{B}^{0}, s_{B}^{0})-comodule.

Definition 5.3. 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 h_{∗}^{B}(S_{B}^{0}, s_{B}^{0}) ≅ h_{∗}(B_{+})-comodule for each i.

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

Suppose {(f_{−i}, s_{−i}), (h_{−i}, u_{−i})} is an h_{∗}^{B}-display of (f, s). Let (g, t) be another pointed space over B. By Lemma 4.7, smashing with (g, t) preserves cofibrations. Thus

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

are cofibrations. Thus we obtain an exact couple by applying h_{∗}^{B}(−). Definition 5.4. Define

D_{−p,q}^{1}((f, s), (g, t)) = h_{−p}^{B}((f_{−p}, s_{−p}) ∧ (g, t))
E_{−p,q}^{1}((f, s), (g, t)) = h_{−p}^{B}((h_{−p}, u_{−p}) ∧ (g, t)).

The cobar spectral sequence or the Künneth spectral sequence defined by an h_{∗}^{B}-display

{(f_{−i}, s_{−i}), (h_{−i}, u_{−i})}

is the spectral sequence associated with this exact couple and is denoted by {E_{∗,∗}^{r}((f, s), (g, t))}.

In order to identify the E^{2}-term, it is important to use a special kind of display. To see this, let (f, s), (g, t) ∈ (Spaces /B)_{∗} and

{(f_{−i}, s_{−i}), (h_{−i}, u_{−i})}

be an h_{∗}^{B}-display. Since h_{∗}^{B}(h_{−i}, u_{−i}) is flat over h_{∗},

h̃_{∗}(Φ(h_{−i}, u_{−i}) ∧ Φ(g, t)) ≅ h̃_{∗}(Φ(h_{−i}, u_{−i})) ⊗_{h_{∗}} h̃_{∗}(Φ(g, t)) = h_{∗}^{B}(h_{−i}, u_{−i}) ⊗_{h_{∗}} h_{∗}^{B}(g, t). Note that the following composition is trivial

h_{∗}^{B}((h_{−i}, u_{−i}) ∧ (g, t)) ⟶ h̃_{∗}(Φ(h_{−i}, u_{−i}) ∧ Φ(g, t))
≅  h_{∗}^{B}(h_{−i}, u_{−i}) ⊗_{h_{∗}} h_{∗}^{B}(g, t)
⟶^{ψ⊗id−id⊗φ} h_{∗}^{B}(h_{−i}, u_{−i}) ⊗_{h_{∗}} h_{∗}^{B}(S_{B}^{0}, s_{B}^{0}) ⊗_{h_{∗}} h_{∗}^{B}(g, t).

where

Δ_{∗} : h_{∗}^{B}(S_{B}^{0}, s_{B}^{0}) ⟶ h_{∗}^{B}(S_{B}^{0}, s_{B}^{0}) ⊗_{h_{∗}} h_{∗}^{B}(S_{B}^{0}, s_{B}^{0})
ψ : h_{∗}^{B}(h_{−i}, u_{−i}) ⟶ h_{∗}^{B}(h_{−i}, u_{−i}) ⊗_{h_{∗}} h_{∗}^{B}(S_{B}^{0}, s_{B}^{0})
φ : h_{∗}^{B}(g, t) ⟶ h_{∗}^{B}(S_{B}^{0}, s_{B}^{0}) ⊗_{h_{∗}} h_{∗}^{B}(g, t)

are coalgebra and comodule structure maps, respectively. Thus we obtain a map

h_{∗}^{B}((h_{−i}, u_{−i}) ∧ (g, t)) ⟶ Ker(ψ ⊗ id − id ⊗ φ).

By the definition of cotensor product

Ker(ψ ⊗ id − id ⊗ φ) = h_{∗}^{B}(h_{−i}, u_{−i})□_{h_{∗}^{B}(S_{B}^{0},s_{B}^{0})}h_{∗}^{B}(g, t).

It is convenient to assume the resulting map

Ψ : h_{∗}^{B}((h_{−i}, u_{−i}) ∧ (g, t)) ⟶ h_{∗}^{B}(h_{−i}, u_{−i})□_{h_{∗}^{B}(S_{B}^{0},s_{B}^{0})}h_{∗}^{B}(g, t)

is an isomorphism. Definition 5.5. Let (f, s) and (g, t) be pointed spaces over B. An h_{∗}^{B}-display of (f, s)

{(f_{−i}, s_{−i}), (h_{−i}, u_{−i})}

is said to be injective with respect to (g, t) if

Ψ : h_{∗}^{B}((h_{−i}, u_{−i}) ∧ (g, t)) ⟶ h_{∗}^{B}(h_{−i}, u_{−i})□_{h_{∗}^{B}(S_{B}^{0},s_{B}^{0})}h_{∗}^{B}(g, t)

is an isomorphism.

The following lemma is immediate from the definition. Lemma 5.6. If {E_{∗,∗}^{r}((f, s) ∧ (g, t)), d^{r}} is a cobar spectral sequence defined by an injective h_{∗}^{B}-display of (f, s) with respect to (g, t), then we have

E_{∗,∗}^{2}((f, s) ∧ (g, t)) ≅ Cotor^{h_{∗}^{B}(S_{B}^{0},s_{B}^{0})}(h_{∗}^{B}(f, s), h_{∗}^{B}(g, t))
= Cotor^{h̃_{∗}(B_{+})}(h̃_{∗}(Φ(f, s)), h̃_{∗}(Φ(g, t))).

This lemma suggests that cobar spectral sequences defined by using injective displays are isomorphic from the E^{2}-term on. In fact, this is the case. Theorem 5.7. Let (f, s) and (g, t) be pointed spaces over B. Let

\begin{matrix}\{(f_{−i}, s_{−i}), (h_{−i}, u_{−i})\} \\ \{(f′_{−i}, s′_{−i}), (h′_{−i}, u′_{−i})\} \end{matrix}

be injective h_{∗}^{B}-displays for (f, s) with respect to (g, t). Let {E^{r}} and {′E^{r}} be the cobar spectral sequences defined by the first and the second display, respectively. Then we have an isomorphism of spectral sequences

E^{r} ≅ ′E^{r}

for r ≥ 2. Proof. See pp.119–120 of [13]. Smith proved this fact by finding an intermediate display

{(f̄_{−i}, s̄_{−i}), (h̄_{−i}, ū_{−i})}

and maps

{(f_{−i}, s_{−i}), (h_{−i}, u_{−i})} ⟶ {(f̄_{−i}, s̄_{−i}), (h̄_{−i}, ū_{−i})} ⟵ {f′_{−i}, s′_{−i}), (h′_{−i}, u′_{−i})}.

The existence of such a display in our case is essentially proved in pp.112–113 of the same paper.

6 Comparison between Rector’s and Smith’s Construction

In this section, we show that Rector’s construction is a cobar spectral sequence for a certain injective display in (Spaces /B)_{∗} and thus obtain an isomorphism of the spectral sequences of Rector’s and Smith’s. Since Rector’s spectral sequence is isomorphic to the classical Eilenberg-Moore spectral sequence (Theorem 2.3), this implies that Smith’s construction is a generalization of the classical construction.

Let p : X ⟶ B be a fibration. Using the functor

G : Spaces /B ⟶ (Spaces /B)_{∗}

we obtain an object G(p) in (Spaces /B)_{∗}. Let f : Y ⟶ B be a continuous map. In the following we construct a display for G(p)

G(p)  ⟶ (ω_{−1}, u_{−1}) ⟶ (p_{−1}, s_{−1})
(p_{−1}, s_{−1}) ⟶ (ω_{−2}, u_{−2}) ⟶ (p_{−2}, s_{−2})
⋮

so that the filtration on Y ×_{B} X induced by this display is the same as Rector’s cosimplicial construction. Recall that Rector’s spectral sequence is induced from the cofibration sequences

⋮
Ω_{−ℓ+1} ⟶^{φ_{−ℓ}}  \overline{Ω}_{−ℓ} ⟶ Ω_{−ℓ}
⋮

while Smith’s spectral sequence is induced from

Φ(G(f) ∧_{B} G(p))  ⟶ Φ(G(f) ∧_{B} (ω_{−1}, u_{−1})) ⟶ Φ(G(f) ∧_{B} (p_{−1}, s_{−1}))
Φ(G(f) ∧_{B} (p_{−1}, s_{−1})) ⟶ Φ(G(f) ∧_{B} (ω_{−2}, u_{−2})) ⟶ Φ(G(f) ∧_{B} (p_{−2}, s_{−2}))
⋮

But it is not difficult to find (p_{−ℓ}, s_{−ℓ}) and (ω_{−ℓ}, u_{−ℓ}) with

Ω_{−ℓ} ≃ Φ(G(f) ∧_{B} (p_{−ℓ}, s_{−ℓ}))
\overline{Ω}_{−ℓ} ≃ Φ(G(f) ∧_{B} (ω_{−ℓ}, u_{−ℓ})).

It is natural that the process of constructing such a display is very similar to that of the cofibration sequences of Rector’s.

We first define (ω_{−ℓ}, u_{−ℓ}) and then, inductively, (p_{−ℓ}, s_{−ℓ}).

Definition 6.1. For ℓ ≥ 0, define

T (ω_{−ℓ}) = B × (B^{ℓ} × X/ Im δ^{1} ∪ ⋯ ∪ Im δ^{ℓ})

where the maps δ_{i} for i = 1, ⋯, ℓ are the maps in the geometric cobar construction with Y = ∗.

ω_{−ℓ} : T (ω_{−ℓ}) ⟶ B

is just the projection onto the first factor. The section

u_{−ℓ} : B ⟶ T (ω_{−ℓ})

is defined by u_{−ℓ}(b) = (b, ∗).

As is the case of the geometric cobar construction, “δ^{0}” induces a map

ψ_{−ℓ} : (ω_{−ℓ}, u_{−ℓ}) ⟶ (ω_{−ℓ−1}, u_{−ℓ−1})

defined by

ψ_{−ℓ}(b; b_{1}, ⋯, b_{ℓ}, x) = (b; b, b_{1}, ⋯, b_{ℓ}, x).

We also have ψ_{−ℓ−1}ψ_{−ℓ} = ∗.

We need to check Lemma 6.2. Suppose f : Y ⟶ B is surjective. Then for ℓ ≥ 0, we have a homeomorphism

\overline{Ω}_{−ℓ} ≅ Φ(G(f) ∧ (ω_{−ℓ}, u_{−ℓ}))

making the following diagram commutative.

[figure: \overline{Ω}_{−ℓ}; Φ(G(f) ∧ (ω_{−ℓ}, u_{−ℓ})); \overline{Ω}_{−ℓ−1}; Φ(G(f) ∧ (ω_{−ℓ−1}, u_{−ℓ−1})); ψ_{−ℓ}; ≅; Φ(1∧ψ_{−ℓ}); ≅]  (5)

Proof. By the definition of the smash product in Spaces_{∗} /B,

T (G(f) ∧ (ω_{−ℓ}, u_{−ℓ})) = T (G(f)) ×_{B} T (ω_{−ℓ})/(y, u_{−ℓ}(b)) ∼ (b, x) if f(y) = b)
= (Y ⨿ B) ×_{B} B × (B^{ℓ} × X/ Im δ^{1} ∪ ⋯ ∪ Im δ^{ℓ})/(y, f(y), ∗) ∼ (f(y), x)
= Y × (B^{ℓ} × X/ Im δ^{1} ∪ ⋯ ∪ Im δ^{ℓ})
⨿B × (B^{ℓ} × X/ Im δ^{1} ∪ ⋯ ∪ Im δ^{ℓ})/(y, ∗) ∼ (f(y), x).

Thus

Φ(T (G(f) ∧ (ω_{−ℓ}, u_{−ℓ})))
= \frac{Y × (B^{ℓ} × X/ Im δ^{1} ∪ ⋯ ∪ Im δ^{ℓ}) ⨿ B × (B^{ℓ} × X/ Im δ^{1} ∪ ⋯ ∪ Im δ^{ℓ})}{(y, ∗) ∼ (f(y), x)}/ B × ∗
= Y × (B^{ℓ} × X/ Im δ^{1} ∪ ⋯ ∪ Im δ^{ℓ})/Y × ∗
= Y × B^{ℓ} × X/ Im δ^{1} ∪ ⋯ ∪ Im δ^{ℓ}
= \overline{Ω}_{−ℓ}.

With this description of the homeomorphism, it is easy to check the diagram (5) commutes.

We can deform any continuous map f : Y ⟶ B into a surjective map up homotopy. This does not change the homotopy type of the pull-back by f since p is a fibration.

In the following, we always assume that f : Y ⟶ B is surjective.

Corollary 6.3. For each ℓ, if h̃_{∗}(B_{+}) is h_{∗}-flat, we have an isomorphism

h_{∗}^{B}(G(f) ∧ (ω_{−ℓ}, u_{−ℓ})) ≅ h_{∗}^{B}(G(f))□_{h_{∗}^{B}(S_{B}^{0},s_{B}^{0})}h_{∗}^{B}(ω_{−ℓ}, u_{−ℓ}).

Thus the sequence of cofibrations

{(p_{−ℓ+1}, s_{−ℓ+1}) ⟶ (ω_{−ℓ}, u_{−ℓ}) ⟶ (p_{−ℓ}, s_{−ℓ})}_{ℓ=1,2,···}

just constructed is an injective h_{∗}^{B}-display of G(p) with respect to G(f).

Proof. Rector proved in [11] that

h̃_{∗}(\overline{Ω}_{−ℓ}) ≅ h̃_{∗}(Y_{+})□_{h̃_{∗}(B_{+})}h̃_{∗}(Φ(ω_{−ℓ}, u_{−ℓ}))
≅ h_{∗}^{B}(G(f))□_{h_{∗}^{B}(S_{B}^{0},s_{B}^{0})}h_{∗}^{B}(ω_{−ℓ}, u_{−ℓ}).

Therefore the display {(p_{−ℓ}, s_{−ℓ}), (ω_{−ℓ}, u_{−ℓ})} is an injective display.

Let us prove that the spectral sequence induced by the above display is identical to Rector’s spectral sequence.

Let α_{−1} : G(p) ⟶ (ω_{−1}, u_{−1}) be the map in (Spaces /B)_{∗} defined on the total space by the following composition.

T (G(p)) = X ⨿ B ⟶^{p×id_{X}⨿id_{B}} B × X ⨿ B = B × X ⨿ B × {∗} = B × X/ϕ = T (ω_{0}) ⟶^{ψ_{−1}} T (ω_{−1}).

Let (p_{−1}, s_{−1}) be the cofiber of α_{−1}. Since ψ_{−1}ψ_{−2} = ∗, ψ_{−2} induces a map

α_{−2} : (p_{−1}, s_{−1}) ⟶ (ω_{−2}, u_{−2})

with ψ_{−3}α_{−2} = ∗. Inductively on ℓ, we can define (p_{−ℓ+1}, s_{−ℓ+1}) and a map

α_{−ℓ} : (p_{−ℓ+1}, s_{−ℓ+1}) ⟶ (ω_{−ℓ}, u_{−ℓ})

with

ψ_{−ℓ−1}α_{−ℓ} = ∗.

By the commutativity of the diagram (5) and Lemma 4.11, we have an equivalence of cofibrations

[figure: Ω_{−ℓ+1}; Φ(G(f) ∧ (p_{−ℓ+1}, s_{−ℓ+1})); \overline{Ω}_{−ℓ}; Φ(G(f) ∧ (ω_{−ℓ}, u_{−ℓ})); Ω_{−ℓ}; Φ(G(f) ∧ (p_{−ℓ}, s_{−ℓ}))]

Thus we obtain Corollary 6.4. Let

[figure: Y ×_{B} X; X; Y; B; p; f]

be a pull-back diagram.

If h̃_{∗}(B_{+}) is h_{∗}-flat, the spectral sequence induced from Rector’s geometric cobar construction for this pull-back diagram is isomorphic to Smith’s spectral sequence from the E^{2}-term on.

7 The Gravity Filtration on the Space of Little Cubes

In [14], the author constructed a spectral sequence, called the gravity spectral sequence, strongly converging to h_{∗}(Ω^{n}Σ^{n}X) for any homology theory h_{∗}(−), which is of the following form

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

if h_{∗}(−) satisfies certain flatness conditions. The purpose of this section is to show that this spectral sequence is isomorphic to the classical Eilenberg-Moore spectral sequence from the E^{2}-term on.

Before we investigate the gravity spectral sequence, let us consider a more general situation. Let

[figure: F; E; {∗}; B.; p]

be a pull-back diagram of pointed spaces. The construction of the Eilenberg-Moore spectral sequence in §5 is based on the notion of display which is a filtration on G(p) in the category of pointed spaces over B. As we have seen in §5, a display induces a stable decreasing filtration on F, and hence a spectral sequence abutting to the homology of F. In some cases, however, the total space E itself has a stable decreasing filtration which induces a stable decreasing filtration on F. In fact, under some conditions, a decreasing filtration on E defines a display for G(p) which induces the same stable filtration on F as the one induced by the filtration on E. To be more precise, consider the following data and conditions.

1. A pull-back diagram of pointed spaces

[figure: F; E; {∗}; B.; p; ι_{0}]

2. A decreasing filtration on E

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

in which each consecutive inclusion

F_{−q−1}E ⟶^{i_{−q−1}} F_{−q}E

is a pointed cofibration.

3. In the induced decreasing filtration on F

⋯ ⊂ F_{−q−1}F ⊂ F_{−q}F ⊂ ⋯ ⊂ F_{−1}F ⊂ F_{0}F = F,

where F_{−q}F = F_{−q}E ∩ F, each consecutive inclusion is also a pointed cofibration. Proposition 7.1. Given the above data, let p_{−q} : F_{−q}E ⟶ B be the restriction of p on F_{−q}E. If

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. Proof. By the construction, {Σ^{q}G(p_{−q}), Σ^{q}C(G(i_{−q−1}))} is a display. Consider the map of cofibrations

[figure: Φ(G(p_{−q−1})); Φ(G(p_{−q})); C(G(i_{−q−1})); F_{−q−1}E_{+}; F_{−q}E_{+}; F_{−q}E/F_{−q−1}E; ∥; ∥]

Since the left and the midle vertical map are homotopoy equivalence, the induced map in the right is a homotopy equivalence. Thus

Φ(G(p_{−q})) ≃ F_{−q}E/F_{−q−1}E.

Similarly

Φ(G(p_{−q} ∘ i_{−q})) ≃ F_{−q}F/F_{−q−1}F

where i_{−q} : F_{−q}F ⟶ F_{−q}E is the inclusion. On the other hand, since

F_{−q}E ×_{B} {∗} = F_{−q}F,

we have

G(p_{−q}) ∧ G(ι_{0}) = G(p_{−q} ∘ i_{−q}).

Therefore

h_{∗}^{B}(Σ^{q}G(p_{−q}) ∧ G(ι_{0})) = h_{∗}^{B}(Σ^{q}G(p_{−q} ∘ i_{−q}))
= h̃_{∗}(Σ^{q}(F_{−q}F/F_{−q−1}F))
≅ h̃_{∗}(Σ^{q}(F_{−q}E/F_{−q−1}E))□_{h̃_{∗}(B_{+})}h_{∗}
= h_{∗}^{B}(Σ^{q}G(p_{−q}))□_{h_{∗}^{B}(S_{B}^{0},s_{B}^{0})}h_{∗}^{B}(G(∗))

and this proves that {Σ^{q}G(p_{−q}), Σ^{q}C(G(i_{−q−1}))} is an injective h_{∗}^{B}-display of G(p) with respect to G(∗).

Corollary 7.2. Under the assumption in the above proposition, the spectral seqeunce defined by the filtration on F

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

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

F ⟶ E ⟶ B

from the E^{2}-term on.

Let us apply this fact to the path-loop fibration,

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

namely the pull-back diagram

[figure: Ω^{n}Σ^{n}X; P Ω^{n−1}Σ^{n}X; {∗}; Ω^{n−1}Σ^{n}X]  (6)

Instead of directly defining a filtration on P Ω^{n−1}Σ^{n}X, we use the little cube model for this pull-back diagram [8].

Let us first recall the definition of little cubes.

Definition 7.3. A little n-cube c is an affine embedding c = (f_{1}, ⋯, f_{n}) : (−1, 1)^{n} ⟶ (−1, 1)^{n} and each f_{i} only depends on the i-th coordinate of (−1, 1)^{n}. Thus the image of c is “parallel” to (−1, 1)^{n}. The configuration space of j little n-cubes C_{n}(j) is the space of ordered j little n-cubes whose images are disjoint from each other.

Definition 7.4. Let (Y, B) be a pointed NDR pair. We define a subspace E_{n}(j; Y, B) of C_{n}(j) × Y^{j} as follows.

(c_{1}, ⋯, c_{j}; y_{1}, ⋯, y_{j}) ∈ E_{n}(j; Y, B) ⟺ if y_{k} ∉ B then c_{k} can be extended

to the right.

where, by “extend to the right”, we mean the following: for a cube c = (f_{1}, ⋯, f_{n}) : (−1, 1)^{n} ⟶ (−1, 1)^{n}, let f′_{1}(t) be the interval with f′_{1}(−1) = f_{1}(−1) and f′_{1}(1) = 1. Let c̃ = (f′_{1}, f_{2}, ⋯, f_{n}). For c = (c_{1}, ⋯, c_{j}) ∈ C_{n}(j), we say c_{k}, for 1 ≤ k ≤ j, can be extended to the right if the image of c̃_{k} does not intersect with the images of other cubes, Im c̃_{k} ∩ Im c_{l} = ϕ for l ≠ k.

By restricting the defining relation of C_{n}(Y) on ⨿C_{n}(j) ×_{Σ_{j}} Y^{j}, we define

E_{n}(Y, B) = (∐_{j} E_{n}(j; Y, B)/Σ_{j})/ ∼ .

For k > 0, we define

F_{k}E_{n}(Y, B) = Im(∐_{j=1}^{k} E_{n}(j; Y, B)/Σ_{j} ⟶ E_{n}(Y, B))

and

Ẽ_{n}(j; Y, B) = F_{j}E_{n}(Y, B)/F_{j−1}E_{n}(Y, B).

These constructions have the following properties

Theorem 7.5. If X is a pointed space with a nondegenerate base point, we have the following weak homotopy equivalences which are natural with respect to X.

Ω^{n}Σ^{n}X ≃_{w} C_{n}(X)
P Ω^{n−1}Σ^{n}X ≃_{w} E_{n}(CX, X).

Theorem 7.6. Under the same condition as above, the sequence

C_{n}(X) ⟶ E_{n}(CX, X) ⟶ C_{n−1}(ΣX)

is a quasi-fibration which is weakly homotopy equivalent to the path-loop fibration

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

under the weak homotopy equivalences in the above theorem.

Theorem 7.7. Under the same condition as above, we have the following natural stable homotopy equivalences

Ω^{n}Σ^{n}X ≃_{S} ⋁_{j=1}^{∞} C_{n}(j) ∧_{Σ_{j}} X^{∧j}  (7)

E_{n}(CX, X) ≃_{S} ⋁_{j=1}^{∞} Ẽ_{n}(j; CX, X)/Σ_{j}.  (8)

With these theorems, we can replace the pull-back diagram (6) by the pull-back diagram

[figure: ⋁_{j=1}^{∞} C_{n}(j) ∧_{Σ_{j}} X^{∧j}; ⋁_{j=1}^{∞} Ẽ_{n}(j; CX, X)/Σ_{j}; {∗}; ⋁_{j=1}^{∞} C_{n−1}(j) ∧_{Σ_{j}} (ΣX)^{∧j}]

which is stably homotopy equivalent to (6).

In [14], Σ_{j}-equivariant decreasing filtrations on C_{n}(j) and Ẽ_{n}(j; CX, X), called the gravity filtrations, are defined. Thus we have a filtration on

⋁_{j=1}^{∞} Ẽ(j; CX, X)/Σ_{j}

and the induced filtration on

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

It is essentially proved in [14] that these gravity filtrations satisfy the condition in Proposition 7.1, hence, by Corollary 7.2, the gravity spectral sequence in [14] is isomorphic to the classical Eilenberg-Moore spectral sequence from the E^{2}-term on.

In order to see this is the case, let us recall the properties of the gravity filtration. The definition of the gravity filtration on C_{n}(j) is very complicated. Instead of describing the filtration, we record the basic properties of the filtration we need.

Theorem 7.8. There exist decreasing Σ_{j}-equivariant filtrations on C_{n}(j) and Ẽ_{n}(j; CX, X)

\begin{matrix}ϕ = F_{−j−1}C_{n}(j) ⊂ F_{−j}C_{n}(j) ⊂ ⋯ ⊂ F_{−1}C_{n}(j) = F_{0}C_{n}(j) = C_{n}(j) \\ ϕ = F_{−j−1}Ẽ_{n}(j; CX, X) ⊂ F_{−j}Ẽ_{n}(j; CX, X) ⊂ ⋯ ⊂ F_{−1}Ẽ_{n}(j; CX, X) ⊂ F_{0}Ẽ_{n}(j; CX, X) \end{matrix}

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. The inclusions

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

are poinited cofibrations for each q.

3. The inclusion

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

Then 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) ∧ Ω^{n−1}Σ^{n}X

for each q. From this theorem it is clear that the gravity filtration on

⋁_{j=1}^{∞} Ẽ_{n}(j; CX, X)/Σ_{j}

satisfies the condition in Proposition 7.1. Corollary 7.9. If h_{∗}(Ω^{n−1}Σ^{n}X) is h_{∗}-flat, the gravity spectral sequence in [14] is isomorphic to the classical Eilenberg-Moore spectral sequence from the E^{2}-term on.

References

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[4] A. Dold and R. Thom. Quasifaserungen und unendliche symmetrische produkte. Ann. of Math., 67:239–281, 1958.

[5] W.G. Dwyer. Exotic convergence of the Eilenberg-Moore spectral sequence. Illinois Journal of Math., 19:607–617, 1975.

[6] S. Eilenberg and J.C. Moore. Homological algebra and fibrations. In Colloque de Topologie (Brussels, 1964). Librairie Universitaire, Louvain, 1966.

[7] S. Eilenberg and J.C. Moore. Homology and fibrations I. Coalgebras, cotensor product and its derived functors. Comment. Math. Helv., 40:199–236, 1966.

[8] J.P. May. The Geometry of Iterated Loop Spaces. Lecture Notes in Mathematics 271. Springer-Verlag, Berlin-Heidelberg-New York, 1972.

[9] R.J. Milgram. The bar construction and abelian H-spaces. Illinois Journal of Math., 11:242–250, 1967.

[10] D.C. Ravenel and W.S. Wilson. The Morava K-theories of Eilenberg-MacLane spaces and the Conner-Floyd conjecture. Amer. J. Math., 102:691–748, 1980.

[11] D.L. Rector. Steenrod operations in the Eilenberg-Moore spectral sequence. Comment. Math. Helv., 45:540–552, 1970.

[12] L. Smith. Lectures on the Eilenberg-Moore Spectral Sequence. Lecture Notes in Mathematics 134. Springer-Verlag, Berlin-Heidelberg-New York, 1970.

[13] L. Smith. On the Künneth theorem. I the Eilenberg-Moore spectral sequence. Math. Z., 116:94–140, 1970.

[14] D. Tamaki. A dual Rothenberg-Steenrod spectral sequence: A construction of a cobar spectral sequence for Ω^{n}Σ^{n}X. preprint, July 1992.