DESUSPENDING AND DELOOPING CELLULAR INEQUALITIES

WOJCIECH CHACHÓLSKI

1. Introduction

This paper is about a certain global approach toward unstable homotopy theory. It has grown out of recent work of A. K. Bousfield (see [2] and [3]) and E. Dror Farjoun (see [7], [8] and [9]), and has been developed by the latter in his recent book [9]. The key roles are played by two geometric relations between spaces which are associated to two ways of “building” one space from another:

Let A be a space.

• By C(A) we denote the smallest class of spaces, which contains A, is closed under weak equivalences, taking homotopy push-outs, and arbitrary wedges. We say that X is A-cellular, or that it is built from A, if X belongs to C(A). We denote this relation by X ≫ A.

• By \overline{C(A)} we denote the smallest class of spaces, which contains C(A), is closed under weak equivalences, and taking extensions by fibrations. That is, for any fibration sequence F → E → B, if F ∈ \overline{C(A)} and B ∈ \overline{C(A)}, then E ∈ \overline{C(A)}. We say that X is A-acyclic, or that X is killed by A, if X belongs to \overline{C(A)}. We denote this relation by X > A.

Following E. Dror Farjoun we will call the relations ≫ and > respectively a strong and a weak cellular inequality.

There are two important functors related to cellular inequalities. They are A. K. Bousfield’s periodization functor P_{A} (see [2, Section 2.8]) and E. Dror Farjoun’s colocalization functor CW_{A} (see [7, Section 3]). A space X is A-acyclic if and only if P_{A}(X) is weakly contractible. Thus the class \overline{C(A)} coincides with the “kernel” of the functor P_{A} (see [5, Theorem 17.3]). Studying classes \overline{C(A)} is the same as studying unstable Bousfield classes (see [2, Section 9]). A space X is A-cellular if and only if the natural map CW_{A}(X) → X is a weak equivalence. Thus the class C(A) coincides with the “image” of the functor CW_{A} (see [5, Theorem 8.2]). This characterization of classes C(A) and \overline{C(A)} gives universal properties that describe A-cellular and A-acyclic spaces (see Theorem 3.7 and Theorem 4.5).

The relations ≫ and > preserve homotopy properties that are invariant under taking homotopy push-outs, arbitrary wedges, and in the case of the relation > extensions by fibrations. Being acyclic with respect to some homology theory, or ptorsion, or n-connected are just some of the examples of these properties. By choosing one of them, using the relations ≫ and >, we can approximate one space by another, where we regard two spaces as being close to each other if they satisfy the chosen property. We will use these relations to prove the following three theorems:

Theorem 1. Let F → E → A be a fibration sequence over a connected space A and h_{∗} be a homology theory. If F → E is an h_{∗}-equivalence, then A is h_{∗}-acyclic (see Corollary 10.3).

The second theorem deals with the question: Which spaces A satisfy the following property: Property (⋆). If A → X is a map such that X is connected, then the following conditions about a space Y are equivalent:

• map_{∗}(A, Y) is weakly contractible.

• map_{∗}(ΩΣA, Y) is weakly contractible.

• map_{∗}(F ib(X → X/A), Y ) is weakly contractible.

The following theorem says that if A is a suspension, then for any map A → X, such that X is connected, the homotopy fiber F ib(X → X/A) and A have unstably quite a lot in common: Theorem 2.

• If A is weakly equivalent to a suspension, then it satisfies the property (⋆) (see Corollary 8.11).

• Let A → Z be a map such that Z is connected. If A satisfies the property (⋆), then so does F ib(Z → Z/A) (see Corollary 8.7).

It turns out that the above theorems are reflections of a certain duality, where we consider A ≫ (>)X as the “dual” of X ≫ (>)A: Theorem 3.

• If A → X → X/A is a cofibration sequence, such that X is connected, then ΩΣA ≫ F ib(X → X/A) ≫ A (see Proposition 8.1).

• If F → E → A is a fibration over a connected space A, then E/F ≫ ΣΩA and ΣA ≫ Σ(E/F) (see Theorem 10.1).

For example to prove Theorem 1, observe that if h_{∗} is a homology theory and F → E → A is a fibration such that F → E is an h_{∗}-equivalence, then E/F is h_{∗}-acyclic, and so is Σ(E/F). Therefore since ΣA ≫ Σ(E/F), we get that A is also h_{∗}-acyclic.

Our goal is to establish various rules for operating on the symbols X ≫ A and X > A. Then use these rules in particular cases to show non trivial cellular inequalities. In this paper we are focusing on a question:

When can we desuspend and deloop cellular inequalities?

We say that A allows desuspending (weak) cellular inequalities if ΣA ≫ (>)ΣX implies A ≫ (>)X. We say that a connected space A allows delooping (weak) cellular inequalities if X connected and ΩX ≫ (>)ΩA, implies X ≫ (>)A. We prove various characterizations of these properties. In particular we show:

Theorem 4.

• A space A allows desuspending (weak) cellular inequalities if the class C(A) (\overline{C(A)}) is generated by a loop space (see Theorem 8.4)

• A connected space A allows delooping (weak) cellular inequalities if the class C(A) (\overline{C(A)}) is generated by a suspension (see Theorem 9.7 and Proposition 10.6).

We also prove a result that was originally observed by E. Dror Farjoun (see Corollary 10.7):

Theorem 5. Let A be a connected space. If A allows delooping weak cellular inequalities, then it allows desuspending weak cellular inequalities. Hence for any X and Y, Σ^{n+1}X > Σ^{n}Y is equivalent to ΣX > Y.

Acknowledgment: I would like to thank the referee for his quick response, detailed reports, and many useful suggestions.

2. Notation and some properties of homotopy fibers and cofibers

This paper is written simplicially. By a space we mean a simplicial set.

The homotopy cofiber of a map A → X is denoted either by X/A if it is clear which map we are considering, or by Cof(A → X) if we want to point out the map.

Let us choose a basepoint in X. The homotopy fiber of a map A → X at the chosen basepoint is denoted by F ib(A → X). If X is connected, then the homotopy type of F ib(A → X) does not depend on the choice of a basepoint in X.

By ΣX we denote the homotopy cofiber of the map X → ∗. If we choose a basepoint in X, by ΩX we denote the homotopy fiber of the basepoint map ∗ → X.

Let B be a pointed space. A sequence F → E → B is called a fibration sequence if the map F → E is weakly equivalent to the map F ib(E → B) → E. The map E → B is called a principal fibration, if there exists a connected space K, such that the sequence E → B → K is a fibration sequence.

Let X be a pointed space. The quotient space (X × Y)/({x_{0}} × Y) is denoted by X ⋊ Y. Analogously if Y is pointed, the quotient space (X × Y)/(X × {y_{0}}) is denoted by X ⋉ Y.

Let X and Y be pointed spaces. The subspace X × {y_{0}} ∪ {x_{0}} × Y of X × Y is denoted by X ∨ Y and is called the wedge of X and Y. The quotient space (X × Y)/(X ∨ Y) is denoted by X ∧ Y and is called the smash of X and Y. For any choice of basepoints in X and S^{1}, ΣX is weakly equivalent to X ∧ S^{1}.

Let X and Y be spaces. The homotopy push-out hocolim(X ←^{p_{1}} X × Y →^{p_{2}} Y) is denoted by X ∗ Y and is called the join of X and Y. For any choice of basepoints in X and Y, the join X ∗ Y is weakly equivalent to Σ(X ∧ Y).

Throughout the paper we will use the following properties of homotopy fibers and cofibers:

Proposition 2.1.

(1) Let W be a pointed space. If the following is a commutative diagram, then:

\begin{matrix}X & ⟶ & Y \\ ↓ & & ↓ \\ Z & ⟶ & W \end{matrix}  \begin{matrix}F ib(hocolim(Z ← X → Y) → W ) ≃ \\ ≃ hocolim(F ib(Z → W) ← F ib(X → W) → F ib(Y → W)). \end{matrix}

(2) Let (X_{0} → X_{1} → X_{2} → ⋯) be a telescope diagram of pointed spaces. Then:

F ib(X_{0} → hocolim(X_{0} → X_{1} → ⋯)) ≃
≃ hocolim(F ib(X_{0} → X_{1}) → F ib(X_{0} → X_{2}) → ⋯)

(3) Let X →^{f} Y →^{g} Z be two composable maps. There is a fibration sequence:

F ib(X →^{f} Y) → F ib(X →^{gf} Z) → F ib(Y →^{g} Z)

(4) Let X →^{f} Y →^{g} Z be two composable maps. There is a cofibration sequence:

Cof(X →^{f} Y) → Cof(X →^{gf} Z) → Cof(Y →^{g} Z)

Proof of (1). See [9, Appendix HL, page 180].

Proof of (2). See [9, Appendix HL, page 178].

Since the proofs for (3) and (4) are analogous, we are going to present only the details of how to show (4).

Proof of (4). Let us consider the following commutative diagram:

∗ ⟵ ∗ ⟶ ∗  ∗
↑  ↑  ↑  ↑
∗ ⟵ X ⟶^{f} Y  Cof(X →^{f} Y)
↓  ↓^{id}  ↓^{g}  ↓
∗ ⟵ X ⟶^{gf} Z  Cof(X →^{gf} Z)
∗ ⟵ ∗ ⟶ Cof(Y →^{g} Z)

Taking first the homotopy colimit of the columns of this diagram, and then of the row so obtain, we get Cof(Y →^{g} Z). By taking the homotopy colimit in the reverse order, we get the cofiber of the map Cof(X →^{f} Y) → Cof(X →^{gf} Z). Hence the proposition is proven.

The goal of the next two sections is to give a brief introduction to cellular inequalities. We are going to state theorems which will be our main tools. Cellular inequalities were introduced by E. Dror Farjoun (see [8] and [9]). For a detailed discussion of this notion see also [4] and [5].

3. Strong cellular inequalities

Let A be a space.

Definition 3.1. The class C(A) is the smallest class of spaces such that:

• A ∈ C(A).

• If X ∈ C(A) and X ≃ Y, then Y ∈ C(A).

• If X_{i} ∈ C(A), then hocolim(X_{0} ← X_{1} → X_{2}) ∈ C(A).

• If X_{i} ∈ C(A), then for any choice of basepoints in X_{i}, ⋁ X_{i} ∈ C(A).

One can think about the class C(A) as generated by A using two simple operations: taking homotopy push-outs and arbitrary wedges.

Remark 3.2. The class C(A) can also be characterized as follows:

• The class C(A) is the smallest class of spaces that contains A and is closed under weak equivalences, and taking pointed homotopy colimits (see [6, Remark 4.3]).

• The class C(A) is the smallest class of spaces that contains A and is closed under weak equivalence, and taking unpointed homotopy colimits of diagrams indexed by contractible categories (see [1], [6, Corollary 7.7], and [9, Section 2.D.2.7]).

Notation. If X ∈ C(A), then we will write X ≫ A, and call the relation ≫ a cellular inequality. Sometimes we will also refer to the relation ≫ as a strong cellular inequality. If X ≫ A, then we will say that X is built from A, or that X is A-cellular.

The relation X ≫ A says that there exists a recipe which describes how to build X from A using homotopy push-outs and arbitrary wedges. It does not give any explicit procedure to built X from A.

Here are some examples of classes C(A) for various A: Examples

• The class C(S^{0}) consists of all spaces i.e., any space can be built from S^{0}.

• The class C(S^{n}) consists of all (n − 1)-connected spaces.

• Let M(ℤ/p, n) be the Moore space whose the only nontrivial reduced integral homology is ℤ/p in dimension n. The class C(M(ℤ/p, n)) consists of all (n − 1)- connected spaces X such that π_{n}(X) is a group generated by elements of order p, and for i > n, π_{i}(X) is a p-group.

• Let h_{∗} be a homology theory and A be the wedge of representatives of all isomorphism classes of h_{∗}-acyclic simplicial sets whose cardinality is not bigger than h_{∗}(S^{0}). In this case the class C(A) consists of all h_{∗}-acyclic spaces (see [9, Remark 6.A.1.1]). Any space which generates this class is called a universal h_{∗}-acyclic space.

The goal is to establish various rules for operating on the symbol X ≫ A, and then use these rules in particular cases to show non trivial cellular inequalities. The following proposition and theorems give some basic examples and rules for operating on the symbol X ≫ A.

Proposition 3.3.

(1) If X ≫ B and B ≫ A, then X ≫ A.

(2) The class C(A) is closed under taking telescopes, i.e., if X_{i} are A-cellular, then so is the telescope hocolim(X_{0} → X_{1} → ⋯).

(3) The class C(A) is closed under taking homotopy retracts. If X →^{i} A and A →^{r} X are maps, such that the composition ri is homotopic to the identity, then X is A-cellular. In particular ∗ is A-cellular.

(4) ΣA ≫ A .

(5) If A is not connected, then for any n ≥ 0, C(Σ^{n}A) = C(S^{n}).

Proof of (2). By choosing a basepoint in X_{0}, we can make the diagram (X_{0} → X_{1} → ⋯) to be pointed. For any such a choice of basepoints, the telescope hocolim(X_{0} →^{f_{0}} X_{1} →^{f_{1}} ⋯) is weakly equivalent to the homotopy push-out:

hocolim( ⋁ X_{i} ⟵^{id∨id} ⋁ X_{i} ∨ ⋁ X_{i} \underrightarrow{id ∨ ⋁f_{i}} ⋁ X_{i} ∨ ⋁ X_{i})

Therefore if X_{i} are A-cellular, then so is the telescope.

Proof of (3). Notice that X is weakly equivalent to the telescope hocolim(A →^{ir} A →^{ir} ⋯). Hence by (2) X is A-cellular.

Proof of (5). Since A is not connected, S^{0} is a retract of A. If follows that, for every n ≥ 0, S^{n} is a retract of Σ^{n}A. Hence, by (3), S^{n} ≫ Σ^{n}A. On the other hand, Σ^{n}A is S^{n}-cellular since it is (n − 1)-connected. Thus C(S^{n}) = C(Σ^{n}A).

Theorem 3.4.

(1) Let the following be a homotopy push-out square:

\begin{matrix}A & ⟶ & B \\ ↓ & & ↓ \\ C & ⟶ & D \end{matrix}

If B and D are connected, then:

F ib(C → D) ≫ F ib(A → B).

(2) X ≫ ΣA is equivalent to X being connected and ΩX ≫ A.

(3) If X ≫ A, then for any choice of basepoints in A, X, and B, X ∧ B ≫ A ∧ B. In particular:

• We can suspend cellular inequalities. If X ≫ A, then ΣX ≫ ΣA.

• X ∧ B ≫ S^{0} ∧ B ≃ B.

• If X is connected, then for any choice of basepoints in X and B, X ∧B ≫ ΣB.

(4) We can loop cellular inequalities. If A is connected and X ≫ A, then ΩX ≫ ΩA.

(5) Let f : X → Y be a map such that Y is connected. Then:

F ib(Σf : ΣX → ΣY) ≫ Y/X ≫ ΣF ib(f : X → Y)

(6) Let F → E → B be a fibration. If F and E are A-cellular, then so is B.

Proof of (1). If F ib(A → B) is not connected, then any space is F ib(A → B)- cellular (see Proposition 3.3.(5)). Thus in particular F ib(C → D) ≫ F ib(A → B). If F ib(A → B) is connected see [6, Theorem 9.4].

Proof of (2). By the same argument as in (1), if A is not connected, then the theorem holds trivially. If A is connected see [5, Theorem 10.8].

Proof of (3). See [4, Theorem 4.3].

Proof of (4). If A is not simply connected, then ΩA is not connected. Therefore, by Proposition 3.3.(5), the inequality ΩX ≫ ΩA is always true. If A is simply connected see [5, Corollary 10.4].

Proof of (5). See [5, Proposition 4.5 and Proposition 10.5].

Proof of (6). If A is not connected, then the inequality B ≫ A holds trivially. If A is connected see [6, Corollary 6.5].

Corollary 3.5.

(1) ΩΣA ≫ A.

(2) If A is connected, then A ≫ ΣΩA.

(3) C(ΣA) = C(ΣΩΣA).

Proof of (1). Apply Theorem 3.4.(2) to the inequality ΣA ≫ ΣA.

Proof of (2). Apply Theorem 3.4.(5) to the map ∗ → A.

Proof of (3). We have to show that ΣA ≫ ΣΩΣA and ΣΩΣA ≫ ΣA. By Theorem 3.4.(2) the first inequality is equivalent to ΩΣA ≫ ΩΣA, which is obviously true. By suspending ΩΣA ≫ A (see Theorem 3.4.(3)), we get the second inequality ΣΩΣA ≫ ΣA.

Theorem 3.6 ([4, Theorem 6.3]). Let B be a connected space. Let us consider the following commutative diagram:

H [figure: →; ↓] F  Z
↓  ↓
Y ⟶^{l} E ⟶ E/Y
↓^{f}  ↓^{p}  ↓
G ⟶ X ⟶^{g} B ⟶ B/X

where:

• F → E is the homotopy fiber of p : E → B,

• G → X is the homotopy fiber of g : X → B,

• H → Y is the homotopy fiber of the composition g∘f : Y → B,

• E → E/Y is the homotopy cofiber of l : Y → E,

• B → B/X is the homotopy cofiber of g : X → B,

• maps H → F, H → G and E/Y → B/X are induced by l, f, p and g,

• Z → E/Y is the homotopy fiber of E/Y → B/X. Then Z ≫ hocolim(G ← H → F).

Proof. Let us consider the following commutative diagram:

B ⟵^{id} B ⟶^{id} B  B
↑  ↑  ↑  ↑
X ⟵ Y ⟶ E hocolim(X ← Y → E)
↓  ↓^{id}  ↓^{id}  ↓
∗ ⟵ Y ⟶ E  E/Y
B/X ⟵ B ⟶^{id} B

Taking first the homotopy colimit of the columns of this diagram, and then of the so obtained row, we get B/X. By taking the homotopy colimit in the reverse order, we get hocolim(E/Y ← hocolim(X ← Y → E) → B). Hence the following is a homotopy push-out square:

hocolim(X ← Y → E) ⟶ B
↓  ↓
E/Y  ⟶ B/X

According to Proposition 2.1.(1):

F ib(hocolim(X ← Y → E) → B) ≃ hocolim(G ← H → F)

Since B is connected, then so is B/X. Thus, according to Theorem 3.4.(1):

Z = F ib(E/Y → B/X) ≫ hocolim(G ← H → F).

So far we gave a constructive description of the relation ≫. One can also give a universal one. If X ≫ A, then X inherits all the homotopy properties of A which are preserved by homotopy push-outs and arbitrary wedges. For example if A is acyclic with respect to some homology theory, or p-torsion, or n-connected, then so is any A-cellular space. Therefore the property of being A-cellular is the universal one among all the homotopy properties that are satisfied by A and are preserved by homotopy push-outs and arbitrary wedges. The universality of this property can also be expressed as follows:

Theorem 3.7 (E. Dror Farjoun [7, Theorem 3.1]). A space X is A-cellular if and only if for any choice of basepoints in A and X, whenever a map f : Y → Z between pointed Kan simplicial sets induces a weak equivalence between mapping spaces “from A”, f_{∗} : map_{∗}(A, Y) →^{≃} map_{∗}(A, Z), then it also induces a weak equivalence between mapping spaces “from X”, f_{∗} : map_{∗}(X, Y) →^{≃} map_{∗}(X, Z).

Proof. If A is not connected, we have to show that the implication of the theorem is true for any X. Since in this case S^{0} is a retract of A, therefore f : Y → Z is a retract of f_{∗} : map_{∗}(A, Y) → map_{∗}(A, Z). Thus if the latter map is a weak equivalence, then so is the first one. If A is connected, we refer to [5, Theorem 8.2].

Remark 3.8. In the case A is pointed and connected the above universal property says that a space X is A-cellular if the natural map CW_{A}X → X is a weak equivalence, where CW_{A} is E. Dror Farjoun’s colocalization functor (see [5, Section 7] and [7, Section 3.4]). Thus this universal property identifies the class C(A) with the “image” of some idempotent functor.

4. Weak cellular inequalities

Definition 4.1. The class \overline{C(A)} is the smallest class of spaces which contains C(A) and is closed under weak equivalence and taking extensions by fibrations, i.e., if F → E → B is a fibration sequence such that F and B belong to \overline{C(A)}, then so does E.

One can think about the class \overline{C(A)} as generated by A using some simple operations. In addition to taking homotopy push-outs and arbitrary wedges, as it is in the case of C(A), one is also allowed to take extensions by fibrations.

Remark 4.2. One can show that the class \overline{C(A)} is closed under taking unpointed homotopy colimits of diagrams indexed by categories whose classifying spaces belong to \overline{C(A)} (see [6, Corollary 8.3] and compare with Remark 3.2).

Notation. If X ∈ \overline{C(A)}, then we will write X > A and call the relation > a weak cellular inequality. If X > A, then we say that X is killed by A, or that X is A-acyclic.

The relation X > A says that we can build X from A using homotopy push-outs, arbitrary wedges, and extensions by fibrations. Therefore if X ≫ A, then X > A. In general the reverse implication does not hold. There is a striking example involving loops on spheres. For every n, S^{n} > ΩS^{n+1}, but S^{n} ≫ ΩS^{n+1} happens only when S^{n} is an H-space, i.e., when n = 0, 1, 3, 7 (see [5, Corollary 20.13]).

Here are some examples of classes \overline{C(A)} for various A: Examples

• Since the class C(S^{n}) is already closed under taking extension by fibrations \overline{C(S^{n})} = C(S^{n}).

• For every n, \overline{C(ΩS^{n+1})} = C(S^{n}) (see [5, Corollary 20.13]).

• Let M(ℤ/p, n) be the Moore space. The class \overline{C(M(ℤ/p, n))} consists of all (n − 1)- connected spaces X such that, for i ≥ n, π_{i}(X) is a p-group. Thus in this case

C(M(ℤ/p, n)) ⊊ \overline{C(M(ℤ/p, n))}

• Let h_{∗} be a homology theory. The class of all h_{∗}-acyclic spaces is closed under extensions by fibrations. Therefore if A is a universal h_{∗}-acyclic space, then \overline{C(A)} = C(A).

We will use the following properties of the relation >:

Theorem 4.3.

(1) X > ΣA is equivalent to X being connected and ΩX > A (compare with Theorem 3.4.(2)).

(2) Let F → E → B be a fibration sequence. If F ≫ A and B > ΣA, then E ≫ A.

Proof of (1). In the case A is not connected, by Proposition 3.3.(5), the theorem holds trivially. If A is connected see [5, Theorem 18.5].

Proof of (2). If A is not connected, then, as in (1), the theorem obviously follows from Proposition 3.3.(5).

Let A be connected. By D(A) we denote the class of all those spaces X such that, for any fibration sequence Z → W → X, if Z ≫ A, then W ≫ A (see [6, Definition 7.1 and Proposition 8.2]). To prove the theorem we have to show the inclusion \overline{C(ΣA)} ⊂ D(A). Since, by [6, Corollary 8.7], the class D(A) is closed under extensions by fibrations, it is enough to show C(ΣA) ⊂ D(A).

Let Z → W → X be a fibration sequence. If X ≫ ΣA, then ΩX ≫ A (see Theorem 3.4.(2)). Therefore the homotopy fiber of the map Z → W is A-cellular. Thus, according to Theorem 3.4.(6), if Z ≫ A, then W ≫ A. Hence X ∈ D(A). This shows the inclusion C(ΣA) ⊂ D(A).

Corollary 4.4. If B > ΣA, then B ≫ A.

Proof. Apply Theorem 4.3.(2) to the fibration ∗ → B → B.

So far we gave a constructive description of the relation >. One can also give a universal one. If X > A, then X inherits all the homotopy properties of A which are preserved by homotopy push-outs, arbitrary wedges, and extensions by fibrations. For example if A is acyclic with respect to some homology theory, or p-torsion or, nconnected, then so is any A-acyclic space. Therefore the property of being A-acyclic is the universal one among all the homotopy properties that are satisfied by A and are preserved by homotopy push-outs, arbitrary wedges, and extensions by fibrations. The universality of this property can also be expressed as follows:

Theorem 4.5 ([5, Theorem 17.3]). A space X is A-acyclic if and only if for any choice of basepoints in A and X, and for any pointed Kan simplicial set Y, if the mapping space map_{∗}(A, Y) is weakly contractible, then so is map_{∗}(X, Y).

Proof. In the case A is not connected, we have to show that the implication of the theorem is true for all X. Since in this case S^{0} is a retract of A, therefore Y is a retract of map_{∗}(A, Y). Thus if the latter space is weakly contractible, then so is the first one. If A is connected, we refer to [5, Theorem 17.3, Section 18].

Remark 4.6. In the case A is connected, the above universal property says that, a space X is A-acyclic if P_{A}(X) is weakly contractible, where P_{A} is A. K. Bousfield’s periodization functor (see [2, Section 2.8]). Thus this universal property identifies the class \overline{C(A)} with the “kernel” of some idempotent functor (see [5, Theorem 17.3]).

5. Ganea’s theorem

In this section a generalized Ganea’s theorem is going to be presented. Although this theorem has been known, we will present its proof.

Let X_{0} and X_{1} be pointed spaces. By applying Proposition 2.1.(1) to the following commutative diagram:

∗ ⟶ X_{1}
↓  ↓
X_{0} ⟶ X_{0} × X_{1}

we get Ganea’s theorem:

Theorem 5.1 (Ganea). There is a weak equivalence: F ib(X_{0} ∨ X_{1} ↪ X_{0} × X_{1}) ≃ ΩX_{0} ∗ ΩX_{1}.

There is a generalized version of this theorem. In order to give its formulation we have to introduce some notation. Let us assume that we have a family of pointed spaces (X_{0}, X_{1}, …, X_{n}). By i_{k} (0 ≤ k ≤ n) we denote the following inclusion:

i_{k} : ∏_{\begin{matrix}l≠k \\ 0≤l≤n \end{matrix}} X_{l} ↪ ∏_{0≤l≤n} X_{l} , (x_{0}, …, x_{n−1}) ↦ (x_{0}, …, x_{k−1}, ∗, x_{k}, …, x_{n−1})

Let W be the union of all the images of i_{k}. Thus W is a subspace of the product X_{0} × X_{1} × ⋯ × X_{n}. It consists of (n + 1)-tuples (x_{0}, x_{1}, …, x_{n}), where there is at least one index s for which x_{s} = ∗. We will call W the thick wedge of the family (X_{0}, X_{1}, …, X_{n}).

Theorem 5.2. F ib(W ↪ ∏_{0≤l≤n} X_{l}) ≃ ΩX_{0} ∗ ΩX_{1} ∗ ⋯ ∗ ΩX_{n}.

Proof. The proof is by induction on n. For n = 1 we already know that the statement is true.

Let V denotes the thick wedge of the family (X_{0}, X_{1}, …, X_{n−1}). It is clear from the definition that:

W = colim( ∏_{0≤l≤n−1} X_{l} ↩ V ↪ V × X_{n})

where V ↪ V ×X_{n} is the inclusion V ×∗ ↪ V ×X_{n}. Since this map is a cofibration, we have: W ≃ hocolim(∏_{0≤l≤n−1} X_{l} ↩ V ↪ V × X_{n}).

By analyzing the appropriate homotopy fibers and using Proposition 2.1.(1), we get:

F ib(W ↪ ∏_{0≤l≤n} X_{l}) ≃

≃ hocolim(ΩX_{n} ⟵^{pr_{1}} ΩX_{n} × F ib(V ↪ ∏_{0≤l≤n−1} X_{l}) ⟶^{pr_{2}} F ib(V ↪ ∏_{0≤l≤n−1} X_{l}))

By inductive assumption F ib(V ↪ ∏_{0≤l≤n−1} X_{l}) ≃ ΩX_{0}∗ΩX_{1}∗⋯∗ΩX_{n−1}. Therefore F ib(W ↪ ∏_{0≤l≤n} X_{l}) ≃ (ΩX_{0} ∗ ΩX_{1} ∗ ⋯ ∗ ΩX_{n−1}) ∗ ΩX_{n}.

6. I. M. James Construction

Let X be a pointed space. By J(X) we denote the reduced free simplicial monoid generated by X. For every n ≥ 0, there is an n-fold multiplication map X^{n} → J(X), (x_{1}, …, x_{n}) ↦ x_{1}x_{2} ⋯ x_{n}, whose image will be denoted by J_{n}.

Let (X, X, …, X) be the constant family of (n + 1) pointed spaces. The thick wedge of this family will be denoted by W^{(n)}. There is a unique map W^{(n)} → J_{n} such that, for any 0 ≤ k ≤ n, the composition (X^{n} ↪^{i_{k}} W^{(n)} → J_{n}) is equal to the n-fold multiplication map. This map fits into the following push-out diagram:

W^{(n)} ⟶ X^{n+1}
↓  ↓
J_{n} ⟶ J_{n+1}

Since the maps involved are cofibrations, the above diagram is also homotopy pushout. I. M. James showed that if X is connected, then the map X → hocolim(J_{1} → J_{2} → ⋯) is weakly equivalent to the map X → ΩΣX, which is adjoint to the identity ΣX →^{id} ΣX (see [10, Theorem 2.6]).

Proposition 6.1. If X is connected, then:

F ib(J_{n} → J_{n+1}) ≫ (ΩX)^{(n+1)∗} =^{def} \underbrace{ΩX ∗ ΩX ∗ ⋯ ∗ ΩX}_{n+1}

Proof. According to generalized Ganea’s theorem (see Theorem 5.2):

F ib(W^{(n)} ↪ X^{n+1}) ≃ (ΩX)^{(n+1)∗}

Since X is connected, using Theorem 3.4.(1), we can conclude:

F ib(J_{n} → J_{n+1}) ≫ F ib(W^{(n)} ↪ X^{n+1}) ≃ (ΩX)^{(n+1)∗}

7. Generalized Freudenthal Suspension Theorem

Let X be a pointed space. In order to construct the map ΣΩX → X, which is adjoint to the identity map ΩX →^{id} ΩX, one can use the following algorithm:

• take the homotopy pull-back, forget about X, and take the homotopy push-out:

\begin{matrix}ΩX & ⟶ & ∗ \\ ↓ & & ↓ \\ ∗ & ⟶ & X \end{matrix}  \begin{matrix}ΩX & ⟶ & ∗ \\ ↓ & & ↓ \\ ∗ & ⟶ & ΣΩX \end{matrix}

• the natural map ΣΩX ≃ hocolim(∗ ← ΩX → ∗) → X is adjoint to the identity

map ΩX →^{id} ΩX.

Therefore by Proposition 2.1.(1):

Proposition 7.1. F ib(ΣΩX → X) ≃ ΩX ∗ ΩX.

Dually, to construct the map X → ΩΣX, which is adjoint to the identity map ΣX →^{id} ΣX, one can use the following algorithm:

• take the homotopy push-out, forget about X, and take the homotopy pull-back:

\begin{matrix}X & ⟶ & ∗ \\ ↓ & & ↓ \\ ∗ & ⟶ & ΣX \end{matrix}  \begin{matrix}ΩΣX & ⟶ & ∗ \\ ↓ & & ↓ \\ ∗ & ⟶ & ΣX \end{matrix}

• the natural map X → holim(∗ → ΣX ← ∗) ≃ ΩΣX is adjoint to the identity map

ΣX →^{id} ΣX.

In the case of the map ΣΩX → X we could identify its homotopy fiber. In the case of the map X → ΩΣX we will approximate its homotopy fiber using cellular inequalities. This will give us The Generalized Freudenthal Suspension Theorem:

Theorem 7.2. If X is connected, then F ib(X → ΩΣX) ≫ ΩX ∗ ΩX.

Proof. We will use I. M. James model for the map X → ΩΣX. We will prove, by induction on n, that F ib(X → J_{n}) ≫ ΩX ∗ ΩX. (n=1). For n = 1, since the map X → J_{1} is the identity, the statement is obviously true. (n=2). For n = 2, since X = J_{1}, according to Proposition 6.1:

F ib(X → J_{2}) ≃ F ib(J_{1} → J_{2}) ≫ ΩX ∗ ΩX

(n≥2). Let us assume that the theorem is true for n ≥ 2. We will show that F ib(X → J_{n+1}) ≫ ΩX ∗ ΩX. The map X → J_{n+1} can be factored as (X → J_{n} → J_{n+1}), therefore we have the following fibration sequence (see Proposition 2.1.(3)):

F ib(X → J_{n}) → F ib(X → J_{n+1}) → F ib(J_{n} → J_{n+1})

By the inductive assumption, F ib(X → J_{n}) ≫ ΩX ∗ ΩX. According to Proposition 6.1:

F ib(J_{n} → J_{n+1}) ≫ (ΩX)^{(n+1)∗}

Since n ≥ 2, by Theorem 3.4.(3), F ib(J_{n} → J_{n+1}) ≫ ΣΩX ∗ ΩX. It follows, from Theorem 4.3.(2), that F ib(X → J_{n+1}) ≫ ΩX ∗ ΩX.

To conclude the proof observe (see Proposition 2.1.(2)):

F ib(X → ΩΣX) ≃ F ib(X → hocolim(J_{1} → J_{2} → ⋯)) ≃
≃ hocolim(F ib(X → J_{1}) → F ib(X → J_{2}) → ⋯) ≫ ΩX ∗ ΩX

Corollary 7.3. Let Y be a connected space and f : ΣX → Y, g : X → ΩY be a pair of adjoint maps.

• If Y is simply connected, then:

F ib(f : ΣX → Y) > ΣF ib(g : X → ΩY) ∨ ΩY ∗ ΩY

• If F ib(f : ΣX → Y) is connected, then:

F ib(g : X → ΩY) > ΩF ib(f : ΣX → Y) ∨ ΩX ∗ ΩX

(Notice that this corollary gives weak cellular inequalities.)

Proof. In order to construct the adjoint map to g : X → ΩY we have to take its suspension, and then compose it with the map ΣΩY → Y:

(ΣX ⟶^{f} Y) = (ΣX ⟶^{Σg} ΣΩY → Y)

It follows that we have a fibration sequence (see Proposition 2.1.(3)):

F ib(ΣX ⟶^{Σg} ΣΩY) → F ib(ΣX ⟶^{f} Y) → F ib(ΣΩY → Y)

Since Y is simply connected, ΩY is connected, hence by Theorem 3.4.(5):

F ib(Σg : ΣX → ΣΩY) ≫ ΣF ib(g : X → ΩY)

Since F ib(ΣΩY → Y) ≃ ΩY ∗ ΩY, it follows that:

F ib(f : ΣX → Y) > ΣF ib(g : X → ΩY) ∨ ΩY ∗ ΩY

In order to construct the adjoint map to f : ΣX → Y we have to loop this map, and than pre-compose it with the map X → ΩΣX:

(X ⟶^{g} ΩY) = (X → ΩΣX ⟶^{Ωf} ΩY)

Therefore we have a fibration sequence (see Proposition 2.1.(3)):

F ib(X → ΩΣX) → F ib(X ⟶^{g} ΩY) → F ib(ΩΣX ⟶^{Ωf} ΩY)

Since F ib(X → ΩΣX) ≫ ΩX ∗ ΩX (see Theorem 7.2), we get:

F ib(g : X → ΩY) > ΩF ib(f : ΣX → Y) ∨ ΩX ∗ ΩX

Remark 7.4. There is a cellular version of Blakers-Massey theorem (see [4, Theorem 7.1]). It says that if the following is a homotopy push-out square of connected spaces:

\begin{matrix}A & ⟶ & B \\ ↓ & & ↓ \\ C & ⟶ & D \end{matrix}  \begin{matrix}then \\ F ib(ΣA → Σ holim(C → D ← B)) ≫ \\ ≫ F ib(A → B) ∧ F ib(A → C). \end{matrix}

Let us consider the following homotopy push-out square:

\begin{matrix}X & ⟶ & ∗ \\ ↓ & & ↓ \\ ∗ & ⟶ & ΣX \end{matrix}

In this case the above theorem does not give any new information. The map ΣX → ΣΩΣX is weakly equivalent to the inclusion ΣX ↪ ⋁_{n≥1} Σ(X^{n∧}), thus we can even identify its homotopy fiber:

F ib(ΣX → ΣΩΣX) ≃ Ω( ⋁_{n≥2} Σ(X^{n∧}) ⋊ ΩΣX)

Therefore The Generalized Freudenthal Suspension Theorem (Theorem 7.2) can be thought of as a sharper version of the cellular Blakers-Massey theorem. This suggests the following question: Is it true that if the following is a homotopy push-out square of connected spaces:

\begin{matrix}A & ⟶ & B \\ ↓ & & ↓ \\ C & ⟶ & D \end{matrix}  \begin{matrix}then \\ F ib(A → holim(C → D ← B)) ≫ (>) \\ ≫ (>) ΩF ib(A → B) ∗ ΩF ib(A → C) ? \end{matrix}

8. Desuspending cellular inequalities

In this section we are going to discuss under what circumstances we can desuspend cellular inequalities. We start with a preliminary result:

Proposition 8.1. Let A → X → X/A be a cofibration sequence. If X is connected, then:

ΩΣA ≫ F ib(X → X/A) ≫ A

Proof. Consider the following two homotopy push-out squares:

\begin{matrix}A & ⟶ & ∗ \\ ↓ & & ↓ \\ X & ⟶ & X/A \end{matrix}  \begin{matrix}X & ⟶ & X/A \\ ↓ & & ↓ \\ ∗ & ⟶ & ΣA \end{matrix}

By applying Theorem 3.4.(1), we can conclude ΩΣA ≫ F ib(X → X/A) ≫ A.

If we suspend the inequalities stated in the above proposition, then Corollary 3.5.(3) implies:

Corollary 8.2 (E. Dror Farjoun [8]). Let A → X be a map. If X is connected, then C(ΣA) = C(ΣF ib(X → X/A)) = C(ΣΩΣA).

Definition 8.3. We say that a space A allows desuspending (weak) cellular inequalities if ΣA ≫ (>)ΣX implies A ≫ (>)X.

The following theorem shows that the problem of desuspending cellular inequalities is closely related to reversing the inequalities stated in Proposition 8.1:

Theorem 8.4. The following statements are equivalent:

(1) A allows desuspending (weak) cellular inequalities.

(2) A ≫ (>)ΩΣA.

(3) C(A) = C(ΩΣA) (\overline{C(A)} = \overline{C(ΩΣA)}).

(4) If A → X → X/A is a cofibration sequence such that X is connected, then A≫ (>)F ib(X → X/A).

(5) If A → X → X/A is a cofibration sequence such that X is connected, then C(A) = C(F ib(X → X/A)) (\overline{C(A)} = \overline{C(F ib(X → X/A))}).

(6) There exists a connected space B such that C(A) = C(ΩB) (\overline{C(A)} = \overline{C(ΩB)}).

Proof. (1)⇒(2). Notice that, by Theorem 3.4.(2), the inequality ΩΣA ≫ ΩΣA is equivalent to ΣA ≫ ΣΩΣA. Thus since A allows desuspending (weak) cellular inequalities, we get A ≫ (>)ΩΣA. (2)⇔(3)⇔(4)⇔(5)⇒(6). These implications are either obvious or are easy corollaries of Proposition 8.1. (6)⇒(1). Let ΣA ≫ (>)ΣX. We have to show that A ≫ (>)X. Let us consider the following sequence of inequalities:

A ≫ (>)ΩB ≫^{(a)} ΩΣΩB ≫ (>)^{(b)} ΩΣA ≫ (>)^{(c)} X

• By looping the inequality B ≫ ΣΩB (see Corollary 3.5.(2)), we get (a).

• By suspending and looping the inequality ΩB ≫ (>)A, we get (b).

• Since ΣA ≫ (>)ΣX, according to Theorem 3.4.(2) (Theorem 4.3.(1) ), ΩΣA≫ (>)X. Hence we get (c).

Corollary 8.5. If A allows desuspending strong cellular inequalities, then it allows desuspending weak cellular inequalities.

Remark 8.6. Let us consider the following property of a space A: If A → X → X/A is a cofibration sequence such that X is connected, then C(A) = C(F ib(X → X/A)) (\overline{C(A)} = \overline{C(F ib(X → X/A))}).

Theorem 8.4 indicates that this property depends only on the class C(A) (\overline{C(A)}) generated by A, rather than on the homotopy type of A (compare with Remark 9.8 and Remark 10.9).

Theorem 8.4 gives us a simple procedure of constructing new spaces that allow desuspending (weak) cellular inequalities out of old ones:

Corollary 8.7.

• Let B be a connected space. The loop space ΩB allows desuspending cellular inequalities.

• Let A → X → X/A be a cofibration sequence such that X is connected. If we can desuspend (weak) cellular inequalities with A, then so can we with F ib(X → X/A) and ΩΣA.

In order to start the above procedure we would like to have some examples of spaces for which we can desuspend cellular inequalities. If A is a connected H-space, it is a retract of ΩΣA, hence we have a strong cellular inequality A ≫ ΩΣA. Therefore H-spaces allow desuspending cellular inequalities. In general though, it is difficult to expect that there is a strong cellular inequality A ≫ ΩΣA, since according to [5, Corollary 20.13], S^{n} ≫ ΩS^{n+1} if and only if S^{n} is an H-space. However, it is always true that we have a weak cellular inequality S^{n} > ΩS^{n+1}.

It was originally observed by E. Dror Farjoun that in general suspensions do allow desuspending weak cellular inequalities:

Theorem 8.8. If \overline{C(A)} = \overline{C(ΣB)}, then A allows desuspending weak cellular inequalities.

Proof. If A is not simply connected, then B is not connected, and therefore \overline{C(A)} = \overline{C(ΣB)} = C(S^{1}). Since the property of desuspending weak cellular inequalities depends only on the class \overline{C(A)}, and S^{1} has this property, then so does A.

Let A be simply connected. We have to show that A > ΩΣA. Therefore it is enough to prove that F ib(A → ΩΣA) is built from ΩΣA. Using The Generalized Freudenthal Suspension Theorem (Theorem 7.2), we will show more (see Corollary 4.4):

Lemma 8.9. If A is simply connected and \overline{C(A)} = \overline{C(ΣB)}, then F ib(A → ΩΣA) > ΣΩΣA.

Proof. According to Theorem 7.2, F ib(A → ΩΣA) ≫ ΩA ∗ ΩA. Therefore to prove the lemma it is enough to show ΩA ∗ ΩA > ΣΩΣA. Let us consider the following sequence of inequalities:

ΩA ∗ ΩA ≃ ΣΩA ∧ ΩA ≫^{(a)} ΣΣΩA >^{(b)} ΣΣΩΣB ≫^{(c)} ΣΣB >^{(d)} ΣA ≫^{(e)} ΣΩΣA

• Since A is simply connected, then ΩA is connected, and therefore (a) follows from Theorem 3.4.(3)).

• By looping and suspending twice the inequality A > ΣB, we get (b).

• Inequalities (c) and (e) are consequences of Corollary 3.5.(3).

• By suspending the inequality ΣB > A, we get (d).

Corollary 8.10 (E. Dror Farjoun [9, Corollary 5.B.6]). Σ^{n+1}A > Σ^{n}B if and only if ΣA > B.

Corollary 8.11. Let A → X → X/A be a cofibration sequence such that X is connected, Y be a pointed Kan simplicial set, and ξ be any homotopy property of spaces which is preserved under taking homotopy push-outs, arbitrary wedges, and extensions by fibrations. If A ≃ ΣB, then A allows desuspending weak cellular inequalities. Therefore:

\overline{C(A)} = \overline{C(F ib(X → X/A))} = \overline{C(ΩΣA)}

It follows that A satisfies ξ if and only if F ib(X → X/A) does. In particular, for any choice of a basepoint in A, the following statements about Y are equivalent (see Theorem 4.5):

• map_{∗}(A, Y) is weakly contractible.

• map_{∗}(F ib(X → X/A), Y ) is weakly contractible.

• map_{∗}(ΩΣA, Y) is weakly contractible.

In the next two sections the question of delooping cellular inequalities will be addressed. We will discuss the cases of strong and weak cellular inequalities separately.

9. Delooping strong cellular inequalities

We start with a theorem which says that under some rather general assumptions the “dual” statement to Theorem 3.4.(1) is true, where X ≫ A is considered to be the “dual” of A ≫ X.

Theorem 9.1. Let the following be a homotopy pull-back square:

\begin{matrix}Y & ⟶ & E \\ ↓ & & ↓^{p} \\ X & ⟶ & B \end{matrix}

If p is a principal fibration, then B/X ≫ E/Y.

Proof. Let B → K be the classifying map of the principal fibration p : E → B. Thus K is connected. Consider the following commutative diagram:

\begin{matrix}X & ⟶ & B \\ ↓ & & ↓ \\ K & ⟶^{id} & K \end{matrix}

Notice that F ib(B → K) ≃ E, F ib(X → K) ≃ Y, and F ib(K →^{id} K) ≃ ∗. Therefore, according to Theorem 3.6, B/X ≃ F ib(B/X → K/K) ≫ hocolim(∗ ← Y → E) ≃ E/Y.

As a corollary, we get the “dual” statement to Theorem 3.4.(5):

Corollary 9.2. Let X → Y be a map of connected spaces. Then:

Ω(Y/X) ≫ F ib(X → Y) ≫ ΩY/ΩX

Proof. Since Y/X is connected, according to Theorem 3.4.(2), the inequality Y/X ≫ ΣF ib(X → Y) is equivalent to Ω(Y/X) ≫ F ib(X → Y). Thus, by Theorem 3.4.(5), the left inequality of the theorem is true.

Consider the following homotopy pull-back square:

ΩX  → ΩY
↓  ↓
∗ ⟶ F ib(X → Y)

Since X is connected, the map ΩY → F ib(X → Y) is a principal fibration. Therefore, according to Theorem 9.1, F ib(X → Y) ≫ ΩY/ΩX

Corollary 9.3. Let F → E → A be a fibration such that A and E are connected. Then E/F ≫ ΣΩA.

Proof. Let us consider the following homotopy pull-back square:

\begin{matrix}ΩA & ⟶ & ∗ \\ ↓ & & ↓ \\ F & ⟶ & E \end{matrix}

Since E is connected, the map ∗ → E is a principal fibration. Hence we can apply Theorem 9.1, obtaining E/F ≫ ΣΩA.

Corollary 9.4. If F → E → A is a principal fibration such that A and E are connected, then A ≫ E/F ≫ ΣΩA.

Proof. The second inequality is a consequence of Corollary 9.3.

To show the first inequality, consider the following homotopy pull-back square, and apply Theorem 9.1:

\begin{matrix}F & ⟶ & E \\ ↓ & & ↓ \\ ∗ & ⟶ & A \end{matrix}

Remark 9.5. I am not aware of any example of a non principal fibration F → E → A, for which A would not be E/F –cellular or E/F –acyclic. For example, if the map E → A has a section, or A is a suspension, then A is a retract of E/F, and therefore A is E/F -cellular. In the next section we will prove a general, slightly weaker statement that, for any fibration F → E → A, ΣA ≫ Σ(E/F) (see Theorem 10.1).

Definition 9.6. We say that a connected space A allows delooping strong cellular inequalities if X connected and ΩX ≫ ΩA, implies X ≫ A.

The following theorem shows that the problem of delooping cellular inequalities is closely related to reversing the inequalities stated in Corollary 9.4:

Theorem 9.7. Let A be connected. The following statements are equivalent:

(1) A allows delooping strong cellular inequalities.

(2) ΣΩA ≫ A.

(3) C(A) = C(ΣΩA).

(4) If F → E → A is a fibration such that E is connected, then E/F ≫ A.

(5) If F → E → A is a principal fibration such that E is connected, then C(A) = C(E/F).

(6) There exists a space B such that C(A) = C(ΣB).

Proof. (1)⇒(2). By Theorem 3.4.(2), inequality ΣΩA ≫ ΣΩA is equivalent to ΩΣΩA ≫ ΩA. Therefore since A allows delooping cellular inequalities, we get ΣΩA ≫ A. (2)⇔(3)⇔(4) ⇔(5)⇒(6). These implications are either obvious, or are easy consequences of Corollary 9.3 and Corollary 9.4. (6)⇒(1). Let us assume that X is connected and ΩX ≫ ΩA. It follows, from Theorem 3.4.(2), that X ≫ ΣΩA. Therefore, by Corollary 3.5.(3), X ≫ ΣΩA ≫ ΣΩΣB ≫ ΣB ≫ A.

Remark 9.8. Let us consider the following two properties of a connected space A:

(1) If F → E → A is a principal fibration such that E is connected, then C(E/F) = C(A).

(2) If F → E → A is a fibration, not necessarily principal, such that E is connected, then C(E/F) = C(A).

Theorem 9.7 indicates that the property (1) depends only on the class C(A) generated by A, rather than on the homotopy type of A. I do not know whether property (2) also depends only on the class C(A). Observe that if A is a suspension, then for any fibration F → E → A, A is a retract of E/F, and thus A ≫ E/F. In this case, by Theorem 9.7, ΣΩA ≫ A. Therefore suspensions satisfy property (2). If this property depends only on the class C(A), then in the Theorem 9.7 we would not have to restrict ourselves to principal fibrations.

10. Delooping weak cellular inequalities

We start with proving that after only a single suspension the “dual” statement to Proposition 8.1 is also true:

Theorem 10.1. Let A be connected and F → E →^{p} A be a fibration, possibly non principal. Then ΣA ≫ Σ(E/F).

Lemma 10.2. Let f̃: X → X × Y be the graph of the map f : X → Y, i.e., f̃(x) = (x, f(x)). For any choice of a basepoint in Y, Σ((X × Y)/f̃(X)) ≃ Σ(X ⋉ Y). Therefore the suspension of the homotopy cofiber of the graph f̃ depends only on the spaces X and Y, and does not depend on the map f.

Proof. Let us choose a basepoint in Y, and let ∗ : X → Y be the trivial map. Consider the following two sets of composable maps:

X ⟶^{˜∗} X × Y ⟶^{p_{1}} X , X ⟶^{f̃} X × Y ⟶^{p_{1}} X

Notice that both of these compositions equal identity, therefore by Proposition 2.1.(4):

Σ(X ⋉ Y) ≃ Σ((X × Y)/˜∗(X)) ≃ Cof(X × Y ⟶^{p_{1}} X) ≃ Σ((X × Y)/f̃(X))

Proof of the theorem. Consider the following commutative diagram:

E ⟶^{p̃} E × A
↓^{p}  ↓^{p_{2}}
A ⟶^{id} A

According to Theorem 3.6:

(E × A)/p̃(E) ≃ F ib((E × A)/p̃(E) → A/A) ≫ hocolim(∗ ← F → E) ≃ E/F

By suspending this inequality and applying the lemma, we get:

Σ(E ⋉ A) ≃ Σ(E × A)/p̃(E)) ≫ Σ(E/F)

Since A is a retract of E ⋉ A, hence A ≫ E ⋉ A, and thus ΣA ≫ Σ(E/F).

Corollary 10.3. Let F → E → A be a fibration over a connected space A, and h_{∗} be a homology theory. If F → E is h_{∗}-equivalence, then A is h_{∗}-acyclic (notice that we are not making any assumptions about the fibration).

Corollary 10.4. Let F → E → A be a fibration such that A and E are connected. If A allows desuspending (weak) cellular inequalities, then A ≫ (>)E/F ≫ ΣΩA (compare with Corollary 9.4 and Remark 9.5).

Definition 10.5. We say that a connected space A allows delooping weak cellular inequalities if X connected and ΩX > ΩA implies X > A.

Proposition 10.6. Let A be connected. The following statements are equivalent:

(1) A allows delooping weak cellular inequalities.

(2) ΣΩA > A.

(3) \overline{C(A)} = \overline{C(ΣΩA)}.

(4) There exists a space B such that \overline{C(A)} = \overline{C(ΣB)}.

Proof. (1)⇒(2). By Theorem 3.4.(2), ΩΣΩA ≫ ΩA. Since A allows delooping weak cellular inequalities, ΣΩA > A. (2)⇔(3)⇒(4). These implications are either obvious, or are easy corollaries of Theorem 3.4.(5). (4)⇒(1). Let X be connected and ΩX > ΩA. By Theorem 4.3.(1), we get X > ΣΩA > ΣΩΣB > ΣB > A.

Theorem 8.8 together with the above proposition imply:

Corollary 10.7. Let A be connected. If A allows delooping weak cellular inequalities, then it allows desuspending weak cellular inequalities.

Theorem 10.8. Let A be connected. The following statements are equivalent:

(1) A allows delooping weak cellular inequalities.

(2) If F → E → A is a fibration sequence such that E is connected, then E/F > A.

(3) If F → E → A is a fibration sequence such that E is connected, then \overline{C(A)} = \overline{C(E/F)}.

Proof. (1)⇒(2). Since A allows delooping weak cellular inequalities, therefore ΣΩA > A (see Proposition 10.6). Corollary 9.3 implies E/F ≫ ΣΩA > A. (2)⇒(3). We have to show that A > E/F. By Theorem 10.1, ΣA ≫ Σ(E/F). Since A allows delooping weak cellular inequalities, then it also allows desuspending weak cellular inequalities (see Corollary 10.7). Therefore A > E/F. (3)⇒(1). Consider the loop fibration ΩA → ∗ → A. By assumption, \overline{C(A)} = \overline{C(ΣΩA)}, and thus, by Proposition 10.6, A allows delooping weak cellular inequalities.

Remark 10.9. Let us consider the following property of a connected space A: If F → E → A is a fibration such that E is connected, then \overline{C(A)} = \overline{C(E/F)}.

Theorem 10.8 indicates that this property depends only on the class \overline{C(A)} generated by A, rather than on the homotopy type of A (compare with Remarks 8.6 and 9.8).

References

1. A. Amit, Direct limits over diagrams with contractible nerve, Master thesis, Hebrrew Univ. (1993).

2. A. K. Bousfield, Localization and periodicity in unstable homotopy theory, J. Am. Math. Soc. 7(4), 1994, 831-874.

3. A. K. Bousfield, Unstable localization and periodicity, Proc. to the conf. in Alg. Top. Barcelona 1994, Progress in Math. 136, Birkhäuser 1996, 33-50.

4. W. Chachólski, A generalization of the triad theorem of Blakers-Massey, to appear in Topology.

5. W. Chachólski, On the functors CW_{A} and P_{A}, to appear in Duke Journal of Mathematics.

6. W. Chachólski, Closed classes, Proc. to the conf. in Alg. Top. Barcelona 1994, Progress in Math. 136, Birkhäuser 1996, 95-118.

7. E. Dror Farjoun, Cellular spaces, preprint.

8. E. Dror Farjoun, Cellular inequalities, Proc. to the conf. in Alg. Top. Northeastern Univ. June 1993, Springer Verlag.

9. E. Dror Farjoun, Cellular Spaces, Null Spaces and Homotopy Localization, Lecture Notes in Mathematics 1622, Springer Verlag 1996.

10. G. W. Whitehead, Elements of Homotopy Theory, Grad. Texts in Math. 61, Springer Verlag 1978.

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