A GENERALIZATION OF THE TRIAD THEOREM OF BLAKERS-MASSEY

WOJCIECH CHACHÓLSKI

1. Introduction

The purpose of this article is to give some motivating examples for a certain global approach towards unstable homotopy theory. This approach has grown out of recent work of A. K. Bousfield (see [2] and [3]) and E. Dror Farjoun (see [8] and [9]), and has been developed by the latter in his recent book [9]. The key role is played by a geometric relation between spaces which is associated to a process 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. Following E. Dror Farjoun we will call the relation ≫ a cellular inequality.

The relation ≫ preserves homotopy properties that are invariant under taking homotopy push-outs and arbitrary wedges. Being acyclic with respect to some homology theory, or p-torsion, or n-connected are just some of the examples of these properties. By choosing one of them, using the relation ≫, 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 the relation ≫ to unify the following two statements:

Theorem 1.A. Let the following be a homotopy push-out square of connected spaces:

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

(1) (Blakers-Massey [1]). Let A, B, and C be simply connected. If H_{i}(A, B) = ⋆ for i ≤ n (n ≥ 1), and H_{i}(A, C) = ⋆ for i ≤ m (m ≥ 1), then π_{i}(A, B) → π_{i}(C, D) is an isomorphism for i < n + m, and an epimorphism for i ≤ n + m.

(2) Let P and Q be disjoint set of primes. If F ib(A → B) has P -torsion integral homology and F ib(A → C) has Q-torsion integral homology, then the natural map A → holim(C → D ← B) is an integral homology isomorphism.

2 WOJCIECH CHACHÓLSKI

It turns out that these two different statements are reflections of the same geometric phenomena. There is a tautological cellular inequality that generalizes both of them (see Theorem 1.B).

In this article we are addressing the following question:

How far a homotopy push-out square is from being homotopy pull-back? The initial data for our investigation consists of a commutative diagram:

\xymatrix{A \ar[rr] \ar[dd] \ar[dr]^{q} &  & B \ar[dd] \\  & Y \ar[ur] \ar[dl] \\ C \ar[rr] &  & D}

where B and C are connected, and the following are respectively homotopy push-out and homotopy pull-back squares:

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

The map q : A → Y measures how far the above homotopy push-out square is from being homotopy pull-back. In particular, if A → X is a cofibration, we are asking for the difference between A and the homotopy fiber F ib(X → X/A) of the cofiber map X → X/A.

The main theorem of this paper gives a cellular approximation for the homotopy fiber and cofiber of the suspension of the map q : A → Y (see Theorem 9.4 and Corollary 9.5):

Theorem 1.B.

(1) F ib(Σq : ΣA → ΣY) ≫ F ib(A → B) ∧ F ib(A → C).

(2) Σ(Cof(q : A → Y)) ≫ Σ(F ib(A → B) ∧ F ib(A → C)).

Both of the statements of Theorem 1.A are easy consequences of Theorem 1.B (see Corollaries 9.7 and 9.6.(1)). In addition, since Theorem 1.B gives a geometric approximation for the homotopy fiber and cofiber of the map Σq : ΣA → ΣY, we can apply it to make some calculations with generalized homology theories (see Corollaries 8.8 and 9.6).

It should also be stressed that Theorem 1.B is a tautology. No assumption is being made about spaces involved. Therefore it follows that the triad theorem of Blakers-Massey [1] (see Theorem 1.A.(1)), is a corollary of the Hurewicz Isomorphism Theorem [13, Sec. 5, Theorem 4].

2. Notation

In this article 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 F ib(E → B) → E.

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. If X is connected, then for any choice of basepoints in X and S^{1}, ΣX is weakly equivalent to X ∧ S^{1}.

Let K be a simplicial set. By the same symbol K we will also denote the simplex category of K (see [5, Definition 3.1]). Its objects are simplices of K, and morphisms are generated by the arrows σ → d_{i}σ and σ → s_{i}σ, where d_{i}σ is the i-th face of a simplex σ, and s_{i}σ is its i-th degeneracy. These arrows are subject to the usual simplicial relations.

Functors indexed by the category associated with a simplicial set K are called diagrams over this simplicial set. If for every simplex σ, F (σ → s_{i}σ) = id_{F (σ)}, then we say that F is a bounded diagram. If in addition, for any morphism ϕ in K, F (ϕ) is a weak equivalence, then we say that F is a good diagram.

If F : K → Spaces is a bounded diagram, we can form its homotopy colimit ∮_{K}F

(see Definition A.1 and [5, Definition 3.9]). It is a simplicial set, whose set of ndimensional simplices is given by:

(∮_{K}F )_{n} = ∐_{σ∈K_{n}} F (σ)_{n}

The basic properties of this construction are listed in the Appendix A (see also [4] and [5]). The only place where we are going to use this construction is in the proof of Proposition 6.1.

Let h_{∗} be a reduced homology theory. We say that a space X is h_{∗} n-connected if, for i ≤ n, h_{i}(X) = ∗.

4 WOJCIECH CHACHÓLSKI

Let E be the representing spectrum for h_{∗}. We say that h_{∗} has a homotopy unit, if there is a map h : S^{0} → E, such that E ≃ E ∧ S^{0} ⟶^{id∧h} E ∧ E has a left homotopy inverse, i.e., there is r : E ∧ E → E, for which the composition r∘(id ∧ h) is weakly equivalent to id_{E}. For example if a homology theory is represented by a ring spectrum, then it has a homotopy unit. Homology theories with a homotopy unit have the following two properties:

(1) If X is h_{∗} n-connected and Y is h_{∗} m-connected, then X ∧ Y is h_{∗} (n + m + 1)- connected.

(2) Let P and Q be disjoint sets of prime numbers. If h_{∗}(X) is P -torsion and h_{∗}(Y) is Q-torsion, then X ∧ Y is h_{∗}-acyclic.

3. Closed classes and cellular inequalities

In this section we are going to state the definition and give some examples and basic properties of closed classes and cellular inequalities. These notions were introduced by E. Dror Farjoun [9] (see also [5])..

Definition 3.1. A class of spaces C is called closed if:

• It is not empty

• It is closed under weak equivalences, i.e., if X is weakly equivalent to Y and X ∈ C, then Y ∈ C.

• It is closed under taking arbitrary wedges, i.e., if, for i ∈ I, X_{i} ∈ C, then for any choice of basepoints in X_{i}, ⋁_{i∈I} X_{i} ∈ C.

• It is closed under taking homotopy push-outs, i.e., if X_{1} ← X_{2} → X_{3} is a push-out diagram such that, for i = 1, 2, 3, X_{i} ∈ C, then hocolim(X_{1} ← X_{2} → X_{3}) ∈ C.

Example 3.2. Let h_{∗} be a reduced homology theory. The class of all h_{∗} n-connected spaces is closed.

Example 3.3. Let Z be a pointed Kan space. Let C be the class of all spaces X such that, for any choice of a basepoint in X, the mapping space map_{∗}(X, Z) is weakly contractible. The class C is closed.

Example 3.4. Let A be a space. By C(A) we denote the smallest closed class that contains A. The class C(A) can be thought of as generated by A using simple operations: taking weak equivalences, arbitrary wedges,

and homotopy push-outs.

We will use the following notation:

Notation. The elements of the class C(A) will be called A-cellular spaces. The relation X ∈ C(A) will be denoted by X ≫ A and called a cellular inequality.

The relation X ≫ A says that there exists a recipe that describes how to built X from A using homotopy push-outs and arbitrary wedges. 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 following list give some examples of classes C(A), for various A:

• 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. Let 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]).

The following proposition gives some elementary properties of closed classes and cellular inequalities:

Proposition 3.5. Let C be a closed class.

(1) If X ≫ A and A ∈ C, then X ∈ C.

(2) The class C is closed under taking telescopes, i.e., if, for i ≥ 0, X_{i} belongs to C, then so does the telescope hocolim(X_{0} → X_{1} → ⋯).

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

(4) ∗ ∈ C.

(5) ΣA ≫ A.

(6) If A is not connected, then, for every n ≥ 0, C(Σ^{n}A) = C(S^{n}), i.e., for any (n − 1)-connected space X, X ≫ Σ^{n}A.

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 choice of a basepoint, 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, for all i, X_{i} belongs to C, then so does the telescope.

Proof of (3). Notice that X ≃ hocolim(A →^{ir} A →^{ir} ⋯). Hence, by (2), X ∈ C.

Proof of (4). Let A be any element of C. Such a space exists since, by definition, C is not empty. The space ∗ is a retract of A, hence (4) follows from (3).

Proof of (6). Since A is not connected, S^{0} is a retract of A. Thus S^{n} is a retract of Σ^{n}A and (6) follows from (3).

Observe that, by Proposition 3.5.(6), if A is not connected, then X ≫ A holds for any X, i.e., C(A) is the class of all spaces.

4. Smash, join, and cellular inequalities

The purpose of this section is to show that the smash and join constructions preserve cellular inequalities. One should pay attention to the way these statements are proven since often the same idea works in showing other properties of cellular inequalities.

Proposition 4.1. For any choice of a basepoint in B, X ⋉ B ≫ B.

Proof. Let C = { X | for any choice of a basepoint in B, X ⋉ B ≫ B}. We have to show that every space belongs to C. Observe that since S^{0} ⋉ B ≃ B ∨ B ≫ B, thus S^{0} ∈ C. Therefore to show that C is the class of all spaces it is enough to prove that it is a closed class (see Proposition 3.5.(6)). This follows from obvious weak equivalences:

(⋁ X_{i}) ⋉ B ≃ ⋁(X_{i} ⋉ B)

hocolim(X_{0} ← X_{1} → X_{2}) ⋉ B ≃ hocolim(X_{0} ⋉ B ← X_{1} ⋉ B → X_{2} ⋉ B)

Corollary 4.2. For any choice of basepoints in B and X, X ∧ B ≫ B.

Proof. Let us choose basepoints in B and X. The smash product can be expressed as a homotopy push-out X ∧ B ≃ hocolim(∗ ← B → X ⋉ B). All the spaces in this push-out diagram are B-cellular (see Proposition 3.5.(4) and Proposition 4.1), therefore X ∧ B ≫ B.

Theorem 4.3. For any choice of basepoints in A, B, and X, if X ≫ A, then X ∧ B ≫ A ∧ B.

Lemma 4.4. Let B and Z be spaces. The following class is closed:

C = {X | for any choice of basepoints in X and B, X ∧ B ≫ Z}

Proof.

Case 1. Let us assume B ≫ Z. Since X ∧ B ≫ B holds for all X (see Corollary 4.2), thus so does X ∧ B ≫ Z. Hence C is the class of all spaces, and therefore it is closed.

Case 2. Let us assume B ≫̸ Z. We are going to show that C consists of connected spaces. If it is not the case, there is X ∈ C such that X = X_{0} ∐ X_{1}, where X_{0} and X_{1} are not empty. Let us choose a basepoint in X, which lies, for example, in X_{0}. Since, for any choice of a basepoint in B, X ∧ B ≃ (X_{0} ∧ B) ∨ (X_{1} ⋉ B), therefore B is a retract of X ∧ B, and hence B ≫ X ∧ B (see Proposition 3.5.(3)). This implies B ≫ Z, which is a contradiction with our assumption B ≫̸ Z.

For a connected space X, the homotopy type of the smash X ∧ B does not depend on the choice of a basepoint in X. Thus in order to verify that a space X belongs to C, it is enough to check that it is connected and there exists some choice of a basepoint in X such that, for any choice of a basepoint in B, X ∧ B ≫ Z.

Let us consider a push-out diagram X_{0} ← X_{1} → X_{2} of spaces that belongs to C. By choosing a basepoint in X_{1}, we can make this push-out to be a pointed diagram. With this choice of basepoints:

hocolim(X_{0} ← X_{1} → X_{2}) ∧ B ≃ hocolim(X_{0} ∧ B ← X_{1} ∧ B → X_{2} ∧ B)

It follows that C is closed under taking homotopy push-outs.

Similarly, since, with the canonical choice of a basepoint in ⋁ X_{i}, (⋁ X_{i}) ∧ B ≃ ⋁(X_{i} ∧ B), we get that C is also closed under taking arbitrary wedges. Hence C is a closed class.

Proof of the theorem. To prove the theorem let us consider the case Z = A ∧ B. It is clear that A belongs to C. Since C(A) is the smallest closed class that contains A and, by the lemma, C is a closed class, therefore C(A) ⊂ C. Hence if X ≫ A, then for any choice of basepoints in A, B and X, X ∧ B ≫ A ∧ B.

Corollary 4.5.

(1) We can suspend cellular inequalities. If X ≫ A, then ΣX ≫ ΣA.

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

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

Definition 4.6. The join X ∗Y of two spaces X and Y is defined to be the homotopy

push-out X ∗ Y =^{def} hocolim(X ⟵^{p_{1}} X × Y ⟶^{p_{2}} Y).

Observe that to define the join X ∗ Y we do not have to choose basepoints in X and Y.

Although the homotopy type of the smash product X ∧ Y does depend on the choice of basepoints in X and Y, its suspension does not:

Proposition 4.7. For any choice of basepoints in X and Y, X ∗ Y ≃ Σ(X ∧ Y).

8 WOJCIECH CHACHÓLSKI

Proof. Let us choose basepoints in X and Y. Consider the following commutative diagram:

\xymatrix{∗ & X \ar[l] \ar[r]^{id} & X & ∗ \ar[l] \ar[r] & ∗ & ∗ \\ ∗ \ar[u] \ar[d] & X \ar[u]_{id} \ar[d] \ar[l] \ar@{^{(}->}[r] & X × Y \ar[u]_{pr_{1}} \ar[d]^{pr_{2}} & Y \ar[r] \ar@{^{(}->}[l] \ar[d] \ar[u] & ∗ \ar[u] \ar[d] & X ∧ Y \ar[u] \ar[d] \\ ∗ & ∗ \ar[r] \ar[l] & Y & Y \ar[l]_{id} \ar[r] & ∗ & ∗ \\ ∗ & ∗ \ar[l] \ar[r] & X ∗ Y & ∗ \ar[l] \ar[r] & ∗}

Taking first the homotopy colimit of the columns of this diagram, and then of the row so obtain, we get X ∗ Y. By taking the homotopy colimit in the reverse order, we get Σ(X ∧ Y). Therefore X ∗ Y ≃ Σ(X ∧ Y).

Corollary 4.8.

(1) Let h_{∗} be a reduced homology theory with a homotopy unit (see Section 2). If X is h_{∗} n-connected and Y is h_{∗} m-connected, then X ∗Y is h_{∗} (n+m+2)-connected.

(2) If X ≫ A and Y ≫ B, then X ∗ Y ≫ A ∗ B.

5. Homotopy fibers and cellular inequalities

In this section we are going to use cellular inequalities to study the homotopy fiber construction and its behavior with respect to homotopy colimit type of operations. In the following theorems, propositions, and corollaries the role of the connectivity assumptions is to avoid the issue of basepoints. Under these assumptions the homotopy types of the appropriate homotopy fibers do not depend on the choice of basepoints.

Theorem 5.1.

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

(2) We can loop cellular inequalities. Let A be connected. If X ≫ A, then ΩX ≫ ΩA.

Proof of (1). If A is not connected, then the theorem follows trivially from Proposition 3.5.(6). If A is connected see [6, Theorem 9.4].

Proof of (2). If A is not simply connected, then, by Proposition 3.5.(6), the inequality ΩX ≫ ΩA is alway true. If A is simply connected see [6, Corollary 10.4]

One of the most useful tools to study closed classes is The Generalized E. Dror Farjoun Theorem:

Theorem 5.2. Let K be a simplicial set, E : K → Spaces, B : K → Spaces be bounded diagrams, and Ψ : E → B be a natural transformation. If, for every simplex σ ∈ K and for any choice of a basepoint in B(σ), F ib(Ψ_{σ} : E(σ) → B(σ)) belongs to a closed

class C, then, for any choice of a basepoint in ∮_{K}B, F ib(Ψ : ∮_{K}E → ∮_{K}B) ∈ C.

Proof. If C contains a space which is not connected, then, by Proposition 3.5.(6), C contains all spaces. Thus in this case the statement of the theorem becomes trivial. If C consists of connected spaces only, see [5, Theorem 9.1].

Push-out diagrams are of particular importance:

Theorem 5.3. Let the following be a map between homotopy push-out diagrams:

E ≃ hocolim ( E_{1} ⟵ E_{2} ⟶ E_{3} )
↓ ↓ ↓ ↓
B ≃ hocolim ( B_{1} ⟵ B_{2} ⟶ B_{3} )

If, for i = 1, 2, 3 and for any choice of a basepoint in B_{i}, F ib(E_{i} → B_{i}) ≫ A, then, for any choice of a basepoint in B, F ib(E → B) ≫ A.

Corollary 5.4.

(1) Let B be a connected space and the following be a homotopy push-out square:

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

Then for any choice of a basepoint in D

F ib(C → D) ≫ F ib(A → B).

(2) Let A → X → X/A be a cofibration sequence. Then for any choice of a basepoint in X/A, F ib(X → X/A) ≫ A.

Proof of (1). Apply Theorem 5.3 to the following commutative diagram:

C ≃ hocolim ( C ⟵ A ⟶^{id} A )
↓ ↓^{id} ↓^{id} ↓
D ≃ hocolim ( C ⟵ A ⟶ B )

Proof of (2). Apply (1) to the following homotopy push-out square:

A ⟶ P
↓  ↓
X ⟶ X/A

where P is weakly contractible.

In the next proposition we are looking at the behavior of the homotopy fiber with respect to suspending process. In Section 9 we will generalize this proposition to arbitrary homotopy push-outs (see Proposition 9.1).

10 WOJCIECH CHACHÓLSKI

Proposition 5.5. Let f : X → Y be a map.

(1) If Y is connected, then Y/X ≫ ΣF ib(f : X → Y).

(2) F ib(Σf : ΣX → ΣY) ≫ Y/X.

Proof of (1). See [6, Theorem 9.4].

Proof of (2). Apply Corollary 5.4.(2) to the cofibration sequence Y/X → ΣX →^{Σf} ΣY.

Corollary 5.6.

(1) If Y is connected, then Y ≫ ΣΩY.

(2) ΩΣY ≫ Y.

(3) Let h_{∗} be a homology theory, Y be a connected space, and X → Y be a map. If F ib(X → Y) is h_{∗} n-connected, then X/Y is h_{∗} (n + 1)-connected.

Proof of (1) and (2). Apply Proposition 5.5 to any map of the form ∗ → Y.

Proof of (3). Since F ib(X → Y) is h_{∗} n-connected, ΣF ib(X → Y) is h_{∗} (n + 1)- connected. Cellular inequalities preserve h_{∗} m-connectedness for every m, therefore, by Proposition 5.5.(1), Y/X is h_{∗} (n + 1)-connected.

6. Homotopy pull-backs and push-outs

In this section we are going to review some properties of homotopy pull-back and push-out squares. All this material has been known.

Proposition 6.1. Let D be a pointed space. If the following is a commutative diagram:

\begin{matrix}A & ⟶ & B \\ ↓ & & ↓ \\ C & ⟶ & D \end{matrix}  \begin{matrix}then F ib(hocolim(C ← A → B) → D) ≃ \\ ≃ hocolim(F ib(C → D) ← F ib(A → D) → F ib(B → D)). \end{matrix}

Proof. We can assume that D is connected. If not, since we are considering homotopy fibers over the basepoint of D, we can pull-back the commutative square of the proposition over to the basepoint component of D.

By Corollary A.6, we can construct diagrams G : K → Spaces, H : K → Spaces, F : K → Spaces and natural transformations Ψ : H → G, Φ : H → F such that:

• G, H, and F are good diagrams (see Section 2).

• For every simplex simplex σ ∈ K, G(σ), H(σ), and F (σ) are weakly equivalent respectively to F ib(C → D), F ib(A → D), and F ib(B → D).

• The following are weakly equivalent commutative squares:

\begin{matrix}A & ⟶ & B \\ ↓ & & ↓ \\ C & ⟶ & D \end{matrix}  \begin{matrix}∮_{K}H & ⟶^{Φ} & ∮_{K}F \\ ^{Ψ}↓ & & ↓ \\ ∮_{K}G & ⟶ & K \end{matrix}

The homotopy colimit commutes with itself, hence:

hocolim(∮_{K}G ←^{Ψ} ∮_{K}H →^{Φ} ∮_{K}F ) ≃ ∮_{K}hocolim(G ←^{Ψ} H →^{Φ} F)

It follows that the maps hocolim(C ← A → B) → D and ∮_{K}hocolim(G ←^{Ψ} H →^{Φ} F) →

K are weakly equivalent, and so are their homotopy fibers. To conclude the proof, observe that hocolim(G ← H → F) is a good diagram since G, H, and F are. Therefore,

by Proposition A.3, the homotopy fiber of the map ∮_{K}hocolim(G ←^{Ψ} H →^{Φ} F) → K

is weakly equivalent to hocolim(G(σ) ← H(σ) → F (σ)).

Proposition 6.1 gives a methods of verifying whether a commutative square is homotopy push-out:

Corollary 6.2. A commutative square:

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

is homotopy push-out if and only if, for any choice of a basepoint in D, the following space is weakly contractible:

hocolim(F ib(C → D) ← F ib(A → D) → F ib(B → D)) ≃ ∗

Remark. Proposition 6.1 is another formulation of M. Mather’s cube Theorem [10, Theorem 25] (see also [12]). By straightforward application of the criteria given in Corollary 6.2 one can prove:

Theorem.(Mather). Let the following be a commutative diagram:

[figure: A′; B′; C′; D′; A; B; C; D]

If the bottom square is homotopy push-out and the side squares are homotopy pullback, then the tap square is homotopy push-out.

Using Proposition 6.1 we will study how far a homotopy pull-back square if from being homotopy push-out (compare with Sections 8 and 9). The initial data for our investigation consists of a commutative diagram:

\xymatrix{Y \ar[rr] \ar[dd] &  & B \ar[dd] \ar[dl] \\  & Z \ar[dr]^{s} \\ C \ar[rr] \ar[ur] &  & D}

where D is connected, and the following are respectively a homotopy pull-back square and a homotopy push-out square:

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

The map s : Z → D measures how far the above homotopy pull-back square is from being homotopy push-out. We will give a description of its homotopy fiber:

Proposition 6.3. F ib(s : Z → D) ≃ F ib(B → D) ∗ F ib(C → D).

Proof. By possibly changing the diagram in a homotopy meaningful way, we can arrange so that the maps B → D and C → D are fibrations and the following is an “honest” pull-back square:

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

Let G = F ib(C → D), H = F ib(Y → D), and F = F ib(B → D). Since we have an “honest” pull-back square, thus H = F × G, and the induced maps H → F and H → G are the projections. Therefore by Proposition 6.1 and Definition 4.6:

F ib(s : Z → D) ≃ F ib(hocolim(C ← Y → B) → D) ≃ F ∗ G

7. Generalized Serre Theorem

In this section we are going to prove a cellular inequality which generalizes Serre’s result (see [14, Theorem 6.1]).

Theorem 7.1. (Generalized Serre Theorem). Let D be a connected space. Let us consider the following commutative diagram:

H [figure: →; ↓] F  Z
↓  ↓
A ⟶ B  B/A
↓  ↓  ↓
G ⟶ C ⟶ D  D/C

where:

• F → B is the homotopy fiber of B → D,

• G → C is the homotopy fiber of C → D,

• H → A is the homotopy fiber of the composition A → D,

• Z → B/A is the homotopy fiber of B/A → D/C. Then Z ≫ hocolim(G ← H → F).

Proof. Let us consider the following commutative diagram:

D ⟵^{id} D ⟶^{id} D  D
↑  ↑  ↑  ↑
C ⟵ A ⟶ B hocolim(C ← A → B)
↓  ↓^{id}  ↓^{id}  ↓
∗ ⟵ A ⟶ B  B/A
D/C ⟵ D ⟶^{id} D

Taking first the homotopy colimit of the columns of this diagram, and then of the row so obtained, we get D/C. By taking the homotopy colimit in the reverse order,

14 WOJCIECH CHACHÓLSKI

we get hocolim(B/A ← hocolim(C ← A → B) → D). Hence the following is a homotopy push-out square:

hocolim(C ← A → B) ⟶ D
↓  ↓
B/A  ⟶ D/C

According to Proposition 6.1:

F ib(hocolim(C ← A → B) → D) ≃ hocolim(G ← H → F)

Thus, by Corollary 5.4.(1):

Z = F ib(B/A → D/C) ≫ hocolim(G ← H → F).

The case of a homotopy pull-back square is of particular interest:

Corollary 7.2. Let D be a connected space and the following be a homotopy pull-back square:

\begin{matrix}Y & ⟶ & B \\ ↓ & & ↓ \\ C & ⟶ & D \end{matrix}  \begin{matrix}Then \\ F ib(B/Y → D/C) ≫ F ib(C → D) ⋆ F ib(B → D) \end{matrix}

Proof. By changing the homotopy pull-back diagram of the corollary, in a homotopy meaningful way, we can arrange so that the maps C → D, B → D are fibrations and the diagram is a pull-back square. Let G = F ib(C → D), H = F ib(Y → D), and F = F ib(B → D). Since we have an “honest” pull-back square, therefore H = G×F, and the induced maps H → G, H → F are the projections. Thus, according to Theorem 7.1, F ib(B/Y → D/C) ≫ hocolim(F ← H → G) = F ⋆ G.

As a corollary we get a Serre type of Theorem (see [14, Theorem 6.1]) for a generalized homology theory with a homotopy unit:

Corollary 7.3. (Serre). Let D be a connected space and the following be a homotopy pull-back square:

\begin{matrix}Y & ⟶ & B \\ ↓^{g} & & ↓^{f} \\ C & ⟶ & D \end{matrix}

Let h_{∗} be a reduced homology theory with a homotopy unit. If G = F ib(C → D) is h_{∗} n-connected and F = F ib(B → D) is h_{∗} m-connected, then the map h_{i}(f, g) : h_{i}(B/Y) → h_{i}(D/C) is an isomorphism for i ≤ n + m + 2, and an epimorphism for i = n + m + 3.

Proof. The connectivity assumptions on F and G imply that G ∗ H is h_{∗} (n + m + 2)-connected (see Corollary 4.8.(1)). Hence, by Corollary 7.2, F ib(B/Y → D/C) is also h_{∗} (n + m + 2)-connected. It follows that Cof(B/Y → D/C) is h_{∗} (n + m + 3)-connected (see Corollary 5.6.(3)). This is equivalent to the map h_{i}(f, g) : h_{i}(B/Y) → h_{i}(D/C) being an isomorphism for i ≤ n + m + 2 and an epimorphism for i = n + m + 3.

In the following two sections we are going to look into the dual problem to the one analyzed in Section 6. We will consider separately the cases of a cofibration and a general homotopy push-out square.

8. Cofibration and fibration sequences

In this section we are going to apply The Generalized Serre Theorem (see Theorem 7.1) to study the difference between cofibration and fibration sequences (compare with Sections 6 and 9). The line of thought is basically the same as in [11]. The initial data for our investigation consists of a commutative diagram:

\xymatrix{A \ar[rr] \ar[dd] \ar[dr]^{q} &  & P \ar[dd] \\  & F \ar[ur] \ar[dl] \\ X \ar[rr] &  & X/A}

where P is weakly contractible, and the following are respectively homotopy push-out and homotopy pull-back squares:

\begin{matrix}A & ⟶ & P \\ ↓ & & ↓ \\ X & ⟶ & X/A \end{matrix}  \begin{matrix}F & ⟶ & P \\ ↓ & & ↓ \\ X & ⟶ & X/A \end{matrix}

It follows that we have a cofibration sequence A → X → X/A and a fibration sequence F → X → X/A, where F is the homotopy fiber of X → X/A over the component of X/A that contains [A]. The map q : A → F measures the difference between these sequences. Notice that the above data is entirely determined by A → X.

In this section we are going to discuss the suspension of the map q : A → F, which turns out to be just the inclusion of the wedge component (see Corollary 8.2.(2)). We will give a cellular approximation for the homotopy fiber F ib(Σq : ΣA → ΣF), whose cellular type is the same as that of ΩΣ(Cof(q : A → F)) (see Proposition 8.3).

16 WOJCIECH CHACHÓLSKI

Proposition 8.1. The map Σq : ΣA → ΣF has a right homotopy inverse, i.e., there is r : ΣF → ΣA for which the composition ΣA →^{Σq} ΣF →^{r} ΣA is weakly equivalent to the identity ΣA →^{id} ΣA.

Proof. If we take r : ΣF → ΣA to be the map defined by the following commutative diagram, then the proposition clearly follows:

ΣA ≃ hocolim ( ∗ ⟵ A ⟶ ∗ )
↓^{Σq} ↓ ↓^{q} ↓
ΣF [figure: id]  ≃ hocolim ( ∗ ⟵ F ⟶ P )
↓^{r} ↓ ↓ ↓
ΣA ≃ hocolim ( ∗ ⟵ X ⟶ X/A )

Corollary 8.2.

(1) (E. Dror Farjoun [8]). C(ΣA) = C(ΣF).

(2) The map Σq : ΣA → ΣF is weakly equivalent to the inclusion ΣA ↪ ΣA ∨ Σ(F/A).

(3) The map q : A → F induces a monomorphism on any homology.

Proof of (1). We have to show ΣA ≫ ΣF and ΣF ≫ ΣA. According to Corollary 5.4.(2), F >> A. By suspending this inequality (see Corollary 4.5.(1)), we get ΣF >> ΣA. Since ΣA is a retract of ΣF, by Proposition 3.5.(3), ΣA >> ΣF.

Proof of (2). Let us consider the cofibration sequence F/A → ΣA →^{Σq} ΣF. Since Σq has a right homotopy inverse (see Proposition 8.1), the map F/A → ΣA is null homotopic. Hence its cofiber Σq : ΣA → ΣF is weakly equivalent to the inclusion ΣA ↪ ΣA ∨ Σ(F/A).

Proposition 8.3. C(F ib(Σq : ΣA → ΣF)) = C(ΩΣ(F/A)).

Lemma 8.4. Let B be pointed and D be connected and pointed. Then:

C(F ib(B ↪ B ∨ D)) = C(ΩD)

Proof. We start with identifying the homotopy fiber of the inclusion B ↪ B ∨ D. We will show that F ib(B ↪ B ∨ D) ≃ Ω(ΩB ⋉ D).

Consider two pointed maps B ↪ B ∨ D ⟶^{id∨∗} B, whose composition is equal to id_{B}. It follows that there is a fibration sequence:

F ib(B ↪ B ∨ D) → F ib(B →^{id} B) → F ib(B ∨ D ⟶^{id∨∗} B)

Hence F ib(B ↪ B ∨ D) ≃ Ω(F ib(B ∨ D ⟶^{id∨∗} B)).

Using Proposition 6.1, we can identify the homotopy fiber of B ∨ D ⟶^{id∨∗} B as:

F ib(B ∨ D ⟶^{id∨∗} B) ≃ hocolim(F ib(B →^{id} B) ← F ib(∗ → B) → F ib(D →^{∗} B)) ≃

≃ hocolim(∗ ← ΩB ↪ ΩB × D) ≃ ΩB ⋉ D

Therefore F ib(B ↪ B ∨ D) ≃ Ω(ΩB ⋉ D).

According to Proposition 4.1, ΩB ⋉ D ≫ D. Since D is connected, we can loop this inequality (see Theorem 5.1.(2)), obtaining Ω(ΩB ⋉ D) ≫ ΩD. Space D is a retract of ΩB ⋉ D, therefore D ≫ ΩB ⋉ D (see Proposition 3.5.(3)), thus by looping (observe that ΩB ⋉ D is also connected), we get ΩD ≫ Ω(ΩB ⋉ D). This implies C(F ib(B ↪ B ∨ D)) = C(ΩD).

Proof of the proposition. By apply the Lemma to the map Σq : ΣA → ΣF, we get:

C(F ib(Σq : ΣA → ΣF)) = C(F ib(ΣA ↪ ΣA ∨ Σ(F/A))) = C(ΩΣ(F/A))

Here is the main result of this section:

Theorem 8.5. Let A → X be a map such that X/A is connected. Then:

F ib(Σq : ΣA → ΣF) ≫ A ∧ Ω(X/A)

Proof. Let r : ΣF → ΣA be the right homotopy inverse of Σq : ΣA → ΣF constructed in Proposition 8.1.

Step 1. F ib(r : ΣF → ΣA) ≫ F ∗ Ω(X/A).

Let us consider the following homotopy pull-back square:

\begin{matrix}F & ⟶ & P \\ ↓ & & ↓ \\ X & ⟶ & X/ \end{matrix}

Since X/A is connected, according to Corollary 7.2:

F ib(r : ΣF → ΣA) ≃ F ib(P/F → (X/A)/X) ≫ F ∗ Ω(X/A)

Step 2. F ib(r : ΣF → ΣA) ≫ A ∗ Ω(X/A).

By Corollary 5.4, F ≃ F ib(X → X/A) ≫ A. Thus F ∗ Ω(X/A) ≫ A ∗ Ω(X/A) (see Corollary 4.8.(2)) and Step 2 follows from Step 1.

Step 3. F ib(Σq : ΣA → ΣF) ≫ A ∧ Ω(X/A).

The composition ΣA →^{Σq} ΣF →^{r} ΣA is weakly equivalent to the identity, therefore F ib(Σq : ΣA → ΣF) ≃ ΩF ib(r : ΣF → ΣA). Hence by Step 2 and Corollary 5.6.(2):

F ib(Σq : ΣA → ΣF) ≫ Ω(A ∗ Ω(X/A)) ≃ ΩΣ(A ∧ Ω(X/A)) ≫ A ∧ Ω(X/A)

Corollary 8.6.

(1) Let X be connected. Then F ib(Σq : ΣA → ΣF) ≫ A ∧ F ib(A → X).

(2) Let A → X be a map such that X/A is connected. Then:

Σ(F/A) ≫ Σ(A ∧ Ω(X/A)).

(3) Let X be connected. Then Σ(F/A) ≫ Σ(A ∧ F ib(A → X)).

Proof of (1). By Proposition 5.5.(1), X/A ≫ ΣF ib(A → X). It is equivalent to Ω(X/A) ≫ F ib(A → X) (see Theorem 5.1). Therefore Theorems 8.5 and 4.3 imply:

F ib(Σq : ΣA → ΣF) ≫ A ∧ Ω(X/A) ≫ A ∧ F ib(A → X)

Remark. I do not know whether the inequalities of Corollary 8.6 can be, in general, desuspended. For example, if one takes the map A → X to be A → ∗, or A ↪ A∨B, or A ↪ A × B, or S^{n} →^{p} S^{n} for n ≥ 2, then in these cases F/A ≫ A ∧ F ib(A → X).

Under an extra assumption we can get even more explicit formula:

Corollary 8.7. If X/A is weakly equivalent to the suspension of some space, then Σ(F/A) >> A ∧ X/A.

Proof. Let us consider the following sequence of inequalities:

Σ(F/A) >>^{(a)} Σ(A ∧ Ω(X/A)) ≃ A ∧ ΣΩ(X/A) ≫^{(b)} A ∧ X/A

• (a) follows from Corollary 8.6.(2).

• Let X/A ≃ ΣY. By suspending the inequality Ω(X/A) ≃ ΩΣY >> Y (see Corollary 5.6), we get ΣΩ(X/A) >> ΣY ≃ X/A. Hence (b) follows from Theorem 4.3.

Corollaries 8.6.(2) and 8.6.(3) can be used to make homological calculation (compare with Corollary 9.6.(2)).

Corollary 8.8. Let h_{∗} be a reduced homology theory with a homotopy unit (see Section 2). If A is h_{∗} n-connected and either Ω(X/A) or F ib(A → X) is h_{∗} m-connected, then h_{i}(q) : h_{i}(A) → h_{i}(F) is an isomorphism for i ≤ n + m and an epimorphism for i ≤ n + m + 1.

9. Homotopy push-outs and pull-backs

We will start this section with a generalization of Proposition 5.5:

Proposition 9.1. Let f : A → Y be a map and the following be a commutative diagram:

D ≃ hocolim ( B ⟵ A ⟶ C )
↓ ↓^{id} ↓^{f} ↓^{id}
Z ≃ hocolim ( B ⟵ Y ⟶ C )

(1) If Y and Z are connected, then F ib(D → Z) ≫ ΣF ib(f : A → Y).

(2) If Z is connected, then F ib(Σf : ΣA → ΣY) ≫ F ib(D → Z).

The idea of the proof is to decompose the maps D → Z and Σf : ΣA → ΣY into certain homotopy colimits and then apply Theorem 5.2.

Proof of (1). The map D → Z can be expressed as follows:

D ≃ hocolim ( B ← Y → Y ←^{f} A →^{f} Y ← Y → C )
↓ ↓^{id} ↓^{id} ↓^{id} ↓^{f} ↓^{id} ↓^{id} ↓^{id}
Z ≃ hocolim ( B ← Y → Y ← Y → Y ← Y → C )

We can simplify the above diagram by taking the homotopy colimit of the “central” push-out first:

D ≃ hocolim ( B ← Y → X ← Y → C )
↓ ↓^{id} ↓^{id} ↓ ↓^{id} ↓^{id}
Z ≃ hocolim ( B ← Y → Y ← Y → C )

where X = hocolim(Y ← A → Y) and X → Y is the map induced by id_{Y} and f. According to Proposition 6.1, the homotopy fiber F ib(X → Y) is weakly equivalent to the homotopy push-out:

F ib(X → Y) ≃ hocolim(F ib(Y →^{id} Y) ← F ib(A →^{f} Y) → F ib(Y →^{id} Y))

Hence F ib(X → Y) ≃ ΣF ib(f : A → Y). By applying Theorem 5.2 to the above diagram, we get F ib(D → Z) ≫ ΣF ib(f : A → Y).

Proof of (2). The map Σf : ΣA → ΣY can be expressed as follows:

ΣA ≃ hocolim ( ∗ ← B → B ← A → C ← C → ∗ )
↓^{Σf} ↓ ↓^{id} ↓^{id} ↓^{f} ↓^{id} ↓^{id} ↓
ΣY ≃ hocolim ( ∗ ← B → B ← Y → C ← C → ∗ )

We can simplify the above diagram by taking the homotopy colimit of the “central” push-out first:

ΣA ≃ hocolim ( ∗ ← B → D ← C → ∗ )
↓^{Σf} ↓ ↓^{id} ↓ ↓^{id} ↓
ΣY ≃ hocolim ( ∗ ← B → Z ← C → ∗ )

By applying Theorem 5.2 to the above diagram, we get F ib(Σf : ΣA → ΣY) ≫ F ib(D → Z).

In this section we are going to study how far a homotopy push-out square is from being homotopy pull-bac (compare with Sections 6 and 8). The initial data for our

20 WOJCIECH CHACHÓLSKI

investigation consists of a commutative diagram:

\xymatrix{A \ar[rr] \ar[dd] \ar[dr]^{q} &  & B \ar[dd] \\  & Y \ar[ur] \ar[dl] \\ C \ar[rr] &  & D}

where B and C are connected, and the following are respectively homotopy push-out and homotopy pull-back squares:

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

The map q : A → Y measures how far the above homotopy push-out square is from being homotopy pull-back. In this section we are going to investigate the suspension of the map q : A → Y, which turns out to be just the inclusion of the wedge component (see Corollary 9.3.(1)). We will give a cellular approximation for the homotopy fiber F ib(Σq : ΣA → ΣY), whose cellular type is the same as that of ΩΣ(Cof(q : A → Y)) (see Corollary 9.3.(3)).

Proposition 9.2. The map Σq : ΣA → ΣY has a right homotopy inverse, i.e., there is r : ΣY → ΣA for which the composition ΣA →^{Σq} ΣY →^{r} ΣA is weakly equivalent to the identity id_{ΣA}.

Proof. If we take r : ΣY → ΣA to be the map defined by the following commutative diagram, then the proposition clearly follows:

ΣA ≃ hocolim ( ∗ ← B → B ← A → C ← C → ∗ )
↓^{Σq} ↓ ↓^{id} ↓^{id} ↓^{f} ↓^{id} ↓^{id} ↓
ΣY [figure: id]  ≃ hocolim ( ∗ ← B → B ← Y → C ← C → ∗ )
↓^{r} ↓ ↓^{id} ↓^{id} ↓^{f} ↓^{id} ↓^{id} ↓
ΣA ≃ hocolim ( ∗ ← B → D ← D → D ← C → ∗ )

By the same arguments as in Corollary 8.2.(2) and Proposition 8.3, we get:

Corollary 9.3.

(1) The map Σq : ΣA → ΣY is weakly equivalent to ΣA ↪ ΣA ∨ Σ(Y/A).

(2) The map q : A → Y induces a monomorphism on any homology.

(3) C(F ib(Σq : ΣA → ΣY)) = C(ΩΣ(Y/A)).

A GENERALIZATION OF THE TRIAD THEOREM OF BLAKERS-MASSEY 21

Here is the main result of this article: Theorem 9.4. F ib(Σq : ΣA → ΣY) ≫ F ib(A → B) ∧ F ib(A → C). Proof. Let us consider the following commutative diagram:

D ≃ hocolim ( B ⟵ A ⟶ C )
↓ ↓ ↓^{q} ↓
Z [figure: id]  ≃ hocolim ( B ⟵ Y ⟶ C )
↓ ↓ ↓ ↓
D ≃ hocolim ( D ⟵^{id} D ⟶^{id} D )

It follows that F ib(D → Z) ≃ ΩF ib(Z → D). Step 1. F ib(Z → D) ≫ F ib(A → B) ∗ F ib(A → C). Since the following is a homotopy pull-back square:

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

according to Proposition 6.3:

F ib(Z → D) = F ib(hocolim(B ← Y → C) → D) ≃ F ib(B → D) ∗ F ib(C → D) We also have the following homotopy push-out square:

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

According to Corollary 5.4.(1), F ib(B → D) ≫ F ib(A → C) and F ib(C → D) ≫ F ib(A → B). Therefore F ib(Z → D) ≫ F ib(A → B) ∗ F ib(A → C) (see Corollary 4.8.(2)). Step 2. F ib(D → Z) ≫ F ib(A → B) ∧ F ib(A → C). Since F ib(D → Z) ≃ ΩF ib(Z → D), by Step 1:

F ib(D → Z) ≫ Ω(F ib(A → B) ∗ F ib(A → C)) ≃

≃ ΩΣ(F ib(A → B) ∧ F ib(A → C)) ≫ F ib(A → B) ∧ F ib(A → C)

Step 3. F ib(Σq : ΣA → ΣY) ≫ F ib(A → B) ∧ F ib(A → C). According to Proposition 9.1.(2), F ib(Σq : ΣA → ΣY) ≫ F ib(D → Z). Therefore Step 2 implies F ib(Σq : ΣA → ΣY) ≫ F ib(A → B) ∧ F ib(A → C).

22 WOJCIECH CHACHÓLSKI

By applying Proposition 5.5.(1) to the inequality of Theorem 9.4, we get:

Corollary 9.5. Σ(Y/A) ≫ Σ(F ib(A → B) ∧ F ib(A → C)).

Corollary 9.5 can be applied to make homological calculations:

Corollary 9.6. Let h_{∗} be a reduced homology theory with a homotopy unit (see Section 2).

(1) Let P and Q be disjoint sets of prime numbers. If F ib(A → B) has P-torsion h_{∗} homology and F ib(q : A → C) has Q-torsion h_{∗} homology, then q : A → Y is an h_{∗} isomorphism.

(2) If F ib(A → B) is h_{∗} n-connected and F ib(A → C) is h_{∗} m-connected, then h_{i}(q) : h_{i}(A) → h_{i}(Y) is an isomorphism for i ≤ n + m, and an epimorphism for i ≤ n + m + 1.

Using the Hurewicz Isomorphism Theorem (see [13, Sec. 5, Theorem 4]) and Corollary 9.6.(2) in the case of the ordinary integral homology, we can recover the following result of Blakers-Massey (see [14, Theorem 7.14]):

Corollary 9.7. Let the following be a homotopy push-out square of simply connected spaces:

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

If H_{i}(A, B) = ⋆, for i ≤ n (n ≥ 1) and H_{i}(A, C) = ⋆, for i ≤ m (m ≥ 1), then π_{i}(A, B) → π_{i}(C, D) is an isomorphism for i < n + m and an epimorphism for i ≤ n + m.

Appendix A. The homotopy colimit

In this appendix we are going to review the notion of the homotopy colimit of a bounded diagram indexed by a simplicial set. For the proofs of the listed statements we refer to [4] and [5].

We are going to define, by induction on the dimension of a simplicial set K, the

homotopy colimit ∮_{K}F of a bounded diagram F : K → Spaces (see Section 2).

Definition A.1. Let F : K → Spaces be a bounded diagram.

(1) If dim(K) = 0, then ∮_{K}F =^{def} ∐_{σ∈K_{0}} F (σ)

(2) Let τ ∈ (Δ[n])_{n} be the non degenerate simplex of dimension n in Δ[n]. If K = L ∪_{Δ̇[n]} Δ[n], where L is an (n − 1)-dimensional simplicial set, then:

∮_{K}F =^{def} colim(∮_{L}F ← F (τ) × Δ̇[n] ↪ F (τ) × Δ[n])

The main property of this construction is its homotopy invariance, which can be proven by the induction on the dimension of K:

Proposition A.2. Let F : K → Spaces, G : K → Spaces be bounded diagrams and Ψ : F → G be a natural transformation. If for every simplex σ ∈ K, the map

Ψ_{σ} : F (σ) → G(σ) is a weak equivalence, then so is Ψ : ∮_{K}F → ∮_{K}G.

There are two natural maps associated with the construction ∮_{K}F:

∮_{K}F → K , ∮_{K}F → colim_{K}F

Due to the following property one can think about good diagrams (see Section 2) as fibrations:

Proposition A.3. If F : K → Spaces is a good diagram, then F ib(∮_{K}F → K) ≃

F (σ).

Let K be a simplicial set. By sdK we denote the nerve of the simplex category associated with K. Explicitly, an n-dimensional simplex in sdK is a sequence of n composable morphisms in K, i.e., (sdK)_{n} is the set of all sequences of the form

(σ_{0} →^{φ_{0}} σ_{1} →^{φ_{1}} ... →^{φ_{n−1}} σ_{n}).

The construction ∮_{K}F can be used to study maps. Let f : X → K be a map. For

every simplex σ ∈ K, let F_{f}(σ) be the space that fits into the following pull-back square:

F_{f}(σ) ⟶ X
↓  ↓^{f}
Δ[dim(σ)] ⟶_{σ} K

Roughly speaking F_{f}(σ) is the inverse image in X of the simplex σ ∈ K. This construction clearly defines a functor F_{f} : K^{op} → Spaces, where K^{op} denotes the opposite category of the simplex category of K. Out of this functor we can construct a bounded diagram df : sdK → Spaces such that, for an n-dimensional simplex v =

(σ_{0} →^{φ_{0}} σ_{1} →^{φ_{1}} ... →^{φ_{n−1}} σ_{n}) in sdK:

df(v) = F_{f}(σ_{n})

df(d_{i} : v → d_{i}v) = \begin{cases}id : F_{f}(σ_{n}) → F_{f}(σ_{n}) & if i < n \\ F_{f}(φ_{n−1}) : F_{f}(σ_{n}) → F_{f}(σ_{n−1}) & if i = n \end{cases}

The construction d is natural. If f : X → K, g : Y → K and h : X → Y are maps such that g∘h = f, then there is a natural transformation df → dg. In particular there is a natural transformation Ψ : df → d(id) induced by f : X → K itself.

If the map f : X → K is a fibration, then df : sdK → Spaces is a good diagram, whose values are weakly equivalent to the homotopy fibers of f : X → K.

By taking the colimits, out of the diagram df : sdK → Spaces, we can recover back the map:

(colim_{sdK}Ψ : colim_{sdK}df → colim_{sdK}d(id)) = (f : X → K)

The diagram df : sdK → Spaces also carries homotopical information about f:

Theorem A.4. In the following commutative diagram all the horizontal arrows are weak equivalences:

∮_{sdK}df ⟵^{id} ∮_{sdK}df ⟶ colim_{sdK}df = X
↓  ↓  ↓  ↓f
sdK ⟵ ∮_{sdK}d(id) ⟶ colim_{sdK}d(id) = K

It implies:

Corollary A.5. For every map f : X → K, there exist a natural choice of a bounded

diagram F : L → Spaces, such that f is weakly equivalent to ∮_{L}F → L.

Since every map is naturally weakly equivalent to a fibration, we can strengthen the last corollary:

Corollary A.6. For every map f : X → K, there exist a natural choice of a good diagram F : L → Spaces, whose values are weakly equivalent to the homotopy fibers

of f, such that f : X → K is weakly equivalent to ∮_{L}F → L.

References

[1] A. L. Blakers and W. S. Massey, The homotopy group of a triad 1, Ann. of Math. 53, 1951, 161-205.

[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, Homotopy properties of shapes of diagrams, report No. 6, 1993/94, Institut Mittag-Leffler.

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

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