Introduction. and set F = F ib(f) and G = F ib(g). If the total fiber T of the square is connected, then T >

A NEW CELLULAR VERSION OF BLACKERS-MASSEY

WOJCIECH CHACH ´ OLSKI AND J ´ ER ˆ OME SCHERER

Abstract. Consider a push-out diagram of spaces C ← A → B, construct the homotopy push- out, and then the homotopy pull-back of the diagram one gets by forgetting the initial object A. Wwe compare the difference between A and this homotopy pull-back. This difference is measured in terms of the homotopy fibers of the original maps. Restricting our attention to the connectivity of these maps, we recover the classical Blakers-Massey Theorem.

Introduction

This is a very preliminary version. It contains complete proofs, but not all the background and applications! We write F ib(f ) for the homotopy fiber of the map f : A → B. When B is not connected, the homotopy type of these fibers might vary, depending on the chosen base point in B. In the following statement F ib(f ) means then the wedge of all homotopy fibers.

We use the notation X > A when the space X is “killed by A”, i.e. the nullification functor P A

kills X in the sense that P _A X ' ∗. In other words, the space X belongs to the smallest collection of spaces closed under weak equivalences, homotopy colimits, and extensions by fibrations. The coaugmented functor Id → P _A is Bousfield localization with respect to the map A → ∗.

Theorem 5.1. We consider a homotopy push-out square of the form

A ^f //

g



B

 h

C ^k // D

and set F = F ib(f ) and G = F ib(g). If the total fiber T of the square is connected, then T >

ΩF ∗ ΩG.

If P denotes the homotopy pull-back of the diagram C → D ← B, then the total fiber of the square is by definition the homotopy fiber of the canonical map A → P . It thus truly measures the difference between A and P , hence the failure of the homotopy push-out square to be a homo- topy pull-back square. If we focus only on the connectivity of F and G, we obtain the following consequence, which is Goodwillie’s version of the Blakers-Massey Theorem for squares, [5].

2000 Mathematics Subject Classification. Primary 55S45; Secondary 55R15, 55R70, 55P20, 22F50.

1

Theorem 5.1. We consider a homotopy push-out square of the form

A ^f //

g 

B

 h

C ^k // D

If F = F ib(f ) is m-connected and G = F ib(g) is n-connected, then the total fiber is (m + n)- connected.

We can also prove higher dimensional, cubical, analogues of the above theorem, which generalize Goodwillie’s hypercubical statements (but they are not in this note). Our theorem also improves the cellular Blakers-Massey Theorem obtained by the first author [3, Theorem 1.B]. There, the inequality was obtained for the homotopy fiber of the suspension of the map A → P . It is in general impossible to “desuspend” such an inequality, [2], but always easy to suspend ours.

Acknowledgments.

1. Notation and known results

Given a map f : A → B, the homotopy fiber depends on the choice of a base point in B. If we fix such a base point b ∈ B we write F ib b (f ) for the homotopy fiber over b (and its homotopy type only depends on the connected component in which b lies). However we will be mostly interested in global properties of maps and diagrams. Therefore, when we write simply F ib(f ) we mean the collection of all homotopy fibers. From the point of view of cellular and acyclic classes we can think of F ib(f ) as the wedge of F ib _b (f ) where b runs over a set of points b, one in each connected component of B.

Given a commutative square

A //



B



C // D

there is a natural map from A to the homotopy pull-back P of the square of which we removed A, i.e. the diagram B → D ← C. The total fiber, which we often denote by T , is the homotopy fiber of the map A → P . Alternatively it can be constructed as the homotopy fiber of the induced map on horizontal homotopy fibers F ib(A → B) → F ib(C → D).

Remark 1.1. We can always assume that the homotopy push-out is a connected space. If it is

not the case, let us pick a connected component D α ⊂ D and consider the preimages A α ⊂ A,

B _α ⊂ B, and C α ⊂ C of D α . The fibers and total fibers of the homotopy push-out diagram A α //



B α



C α // D α

are some of the fibers and total fibers of the original diagram. Therefore, if the inequality we are looking for holds for this diagram, for any α, then it holds also for the original diagram.

We write X > A, or X ∈ C(A) and say that A kills X when the nullification functor P A “kills”

X, i.e. P A X is contractible. The following lemma is in fact the main defining property of the acyclic classes C(A), namely the closure under extensions by fibrations.

Lemma 1.2. Consider a fibration sequence F → E → B. Then E > F ∨ B.

Recall also that a non-connected space kills all spaces because the inequality X > S ⁰ holds for any space X.

Lemma 1.3. In a homotopy push-out square, F ib(C → D) > F ib(A → B).

Proof. By the previous remark we can suppose that D is connected. Now, if C is not connected, the homotopy fiber F ib(C → D) is not connected either, but since D is connected, there is at least one component in B over which the homotopy fiber is not connected. Therefore the inequality F ib(C → D) > F ib(A → B) holds trivially in this case.

Let us thus assume that C is connected. When B is connected as well, the result has been proven by the first author in [2, Theorem 3.4 (1)]. This allows us to start the (transfinite) induction on the number of components of B. Let us order the components and for any cardinal α call B α ⊂ B the union of the first α components. We can assume that π 0 A ∼ = π 0 B (if not, the claim is obvious) and define D α to be the homotopy push-out of the diagram C ← A α → B α . For a successor β = α + 1 we see that D β is the homotopy push-out of D α ← C → D ⁰ where D ⁰ is the homotopy push-out of connected spaces C ← A ⁰ → B ⁰ and A β = A α ` A <sup>0</sup> and B β = B α ` B ⁰ . We have therefore a fibration

F ib(C → D α ) → F ib(C → D β ) → F ib(D α → D β ).

By induction hypothesis F ib(C → D α ) > F ib(A α → B α ) and from the connected case we deduce that F ib(D α → D β ) > F ib(C → D ⁰ ) > F ib(A ⁰ → B ⁰ ). This proves the claim for D β . When β is a limit ordinal, we conclude by a similar argument using a telescope instead of a push-out.

Here is an example which illustrates both the necessity to deal with non-connected spaces and

the importance to consider all homotopy fibers at once.

Example 1.4. Let f : S ¹ ` S <sup>1</sup> ,→ D <sup>2</sup> ` S ¹ be the disjoint union of the inclusion of the circle as boundary of the disc and the identity. Consider the (homotopy) push-out diagram

S ¹ ` S ¹ //



D ² ` S ¹



S ¹ // D ²

where the left hand-side vertical map is the fold map (the identity on both copies of the circle).

The homotopy pull-back is the disjoint union S ¹ × D ² ` S ¹ × S ¹ and there are thus two total fibers, a point and ΩS ¹ ' Z.

Lemma 1.5. (Chach´ olski, [2, Theorem 7.2]) For any connected space X, the homotopy fiber of the map X → ΩΣX which is adjoint to the identity on ΣX, is killed by ΩX ∗ ΩX.

2. Reduction to a point

Homotopy push-out diagrams in which the terminal object is a point are easier to handle because the homotopy pull-back one needs to form in order to compute the total fiber is simply a product.

The aim of this section is to reduce the Blakers-Massey theorem to this situation. Recall from Remark 1.1 that we can always assume that it is connected.

Given a commutative push-out square

A ^f //

g 

B

 h

C ^k // D

we denote by F the homotopy fiber F ib(f ) and by G the homotopy fiber F ib(g). We will also need the homotopy fibers over D, namely A ⁰ = F ib(A − ^hf − → D), B ⁰ = F ib(h) and C ⁰ = F ib(k).

Proposition 2.1. For any homotopy push-out square as above the commutative square of homotopy fibers over a connected space D:

A ^{0 f}

0

//

g



B ⁰

 C ⁰ // ∗

satisfies the following properties:

• It is a homotopy push-out square;

• F ib(f ⁰ ) ' F and F ib(g ⁰ ) ' G;

• the total fiber T ⁰ = F ib(A ⁰ → B ⁰ × C ⁰ ) is weakly equivalent to T .

Proof. The first statement is a consequence of Mather’s Cube Theorem [6], the second and third fol- low directly from the fact that horizontal (respectively vertical) homotopy fibers agree in homotopy pull-back squares.

Example 2.2. The typical contractible homotopy push-out is that of the diagram A ← A∨B → B where the maps are the obvious collapse maps. Thus the total fiber of the square

A ∨ B //



B

 A // ∗

is the homotopy fiber of the inclusion A ∨ B ,→ A × B. By Puppe’s Theorem [7] this is the join ΩA ∗ ΩB. It is killed by Ω(B o ΩA) ∗ Ω(A o ΩB) = F ib(A ∨ B → A) ∗ F ib(A ∨ B → B) by Lemma 1.3.

Remark 2.3. In the previous example we obtained an estimate on the total fiber of a homotopy push-out square by using the homotopy fibers of the maps to D, before using the ones from A.

This is not to be expected in general as shown by the following example. Let us choose an integral homology equivalence X → Y , for example the one described by Whitehead in [8], where Y = S ¹ and X is obtained from S ¹ ∨ S ² by attaching a single 3-cell via the attaching map

S ² → S ² ∨ S ² ,→

∞

_

−∞

S ² ' ^ S ¹ ∨ S ² → S ¹ ∨ S ²

where the first map has degree 2 on the first sphere and degree −1 on the second one, and the inclusion is on the zeroth and first factors of the infinite wedge. We consider then the homotopy push-out square

X //



S ¹

 ∗ // ∗

The join of the loops of the homotopy fibers of ∗ → ∗ and Spaces ¹ → ∗ is contractible, but the total fiber is the universal cover ˜ X of X, which is not killed by a point.

3. Cofibrations

Given a cofibration A ,→ X, we establish a first estimate for the cellularity of the difference between the space A and the homotopy fiber of X → X/A. Hence we consider in this section a homotopy push-out square of the form

A   //



X



∗ // X/A

and analyze its total fiber T . The homotopy fiber of the map A → X is denoted by F , so that the total fiber T can be identified with the homotopy fiber of the map F → Ω(X/A).

Proposition 3.1. Let A ,→ X → X/A be a cofibration sequence with X connected. Let F be the homotopy fiber of A → X and T the total fiber. Then T > ΩF ∗ ΩF ∨ Ω(F ∗ ΩX) if F is connected.

Proof. The map A → X factors through A/F and we construct a commutative diagram of homo- topy push-out squares

A //



A/F //



X



∗ // ΣF // X/A

Thus the map F → Ω(X/A) factors through the adjoint of the identity F → ΩΣF and we have therefore a fibration sequence

F ib(F → ΩΣF ) → T → F ib(ΩΣF → Ω(X/A)).

Moreover, by Lemma 1.3, F ib(ΣF → X/A) > F ib(A/F → X) and by Lemma 1.5 the first fiber is killed by ΩF ∗ ΩF since we assume here that F is connected. By Ganea’s Theorem [4] F ib(A/F → X) ' F ∗ ΩX and we conclude thus that T > ΩF ∗ ΩF ∨ Ω(F ∗ ΩX).

Corollary 3.2. Let A ,→ X → X/A be a cofibration sequence with X connected. Let F be the homotopy fiber of A → X and T the total fiber. Then T > ΩF ∗ ΩF ∨ Ω(F ∗ ΩX) if T is connected.

Proof. If F is connected the claim has been proven in the previous proposition. If F is not connected, then ΩF contains S ⁰ as a retract. Therefore ΩF ∗ ΩF has the same cellularity as S ⁰ ∗ S ⁰ ' S ¹ . The claim is then true by assumption since we suppose that T is connected.

Example 3.3. When the homotopy fiber F is not connected, we might run into problems if the total fiber is not connected either. This is the case for example for the cofibration sequence S ⁰ ,→ I → S ¹ . Here the homotopy pull-back is ΩS ¹ ' Z and the total fiber consists in the different homotopy fibers of the canonical map S ⁰ → ΩΣS ⁰ . Except two of these fibers which are contractible, they are all empty. This also provides an explanation for the connectivity assumption in Lemma 1.5.

4. A very rough estimate

In this section we obtain a first very rough estimate for the total fiber. By comparing this

apparently useless estimate with our results for cofibration sequences we will be able in the next

section to improve it considerably. We fix now and for the entire section a homotopy push-out

square

A ^f //

g 

B

 C // ∗

and call F = F ib(f ) and G = F ib(g). If b ∈ B, let F b = F ib b (f ). The total fiber T is the homotopy fiber of the map A → B × C, but it can be alternatively described as the homotopy fiber F ib(F → C).

Lemma 4.1. When B and C are connected, the homotopy cofiber C/F of the composite map F → A → C is 2-connected if F is simply connected.

Proof. We have a homotopy push-out square

A/F //



B

 C/F // ∗

Therefore we infer from Lemma 1.3 that C/F = F ib(C/F → ∗) is killed by F ib(A/F → B) = ΩB ∗ F . Since F is simply connected this join is 2-connected.

Here comes the first part of our “very rough estimate”. The roughness of the inequality here comes from the fact that it only involves one of the fibers. As we know from the classical version of the Blakers-Massey Theorem, the connectivity of the total fiber should be related to the sum of the connectivities of both fibers.

Proposition 4.2. Assume that B is connected. If the total fiber T is connected, then T > ΣΩF .

Proof. If F is not simply connected, then ΩF is not connected, and contains thus S ⁰ as a retract.

In this case the proposition only claims that T > ΣS ⁰ = S ¹ , i.e. that T is connected, which is true by assumption.

Let us hence assume that F is simply connected, so that C is simply connected by Lemma 1.3.

By the previous lemma we also know that C/F > S ³ . The total fiber T is the homotopy fiber of F → C and this map factors through H = F ib(C → C/F ). Let us first analyze the homotopy fiber of the map h : F → H. The canonical map F → ΩΣF also factors through h and we have therefore a fibration sequence

F ib(h) → F ib(F → ΩΣF ) → F ib(H → ΩΣF ) .

The total space is killed by ΩF ∗ ΩF ' ΣΩF ∧ ΩF by Lemma 1.5, therefore by ΣΩF . The base space is the total fiber of the homotopy push-out square

C //



C/F



∗ // ΣF

and we know by Proposition 3.1 that it is killed by ΩH ∗ ΩH ∨ Ω(H ∗ Ω(C/F )), since the homotopy fiber of C → C/F is H, which is killed by F by Lemma 1.3 and thus connected. The inequalities H > F and C/F > S ³ imply now that

F ib(H → ΩΣF ) > ΩF ∗ ΩF ∨ Ω(F ∗ S ² ) > Σ ² ΩF ∨ ΩΣ ³ F

where the last inequality comes from our assumption that F be simply connected. Therefore F ib(H → ΩΣF ) > Σ ² ΩF . By the closure property, Lemma 1.2, we infer that F ib(h) > ΣΩF ∨ ΩΣ ² ΩF > ΣΩF .

We finally use a similar trick to get the estimate on T , by considering the composite F → H → C. We obtain here a fibration sequence F ib(h) → T → F ib(H → C). But the base space is Ω(C/F ) and, once again by Lemma 1.3, C/F is killed by F ib(A/F → B) = F ∗ ΩB > ΣF . Thus Ω(C/F ) > ΣΩF and we conclude.

We remove now the connectivity assumption on the space B.

Theorem 4.3. If the total fiber T is connected, then T > ΣΩF , and by symmetry T > ΣΩG.

Proof. We will actually show something a little bit more precise. Just like in the proof of the previous proposition we can assume that F is simply connected, so that C is simply connected as well. Therefore f induces an isomorphism on connected components π 0 A ∼ = π 0 B. Choose a point b in any component and let F b denote the corresponding homotopy fiber F ib b (f ) and T b the corresponding total fiber F ib _(c,b) (A → C × B). We will prove that T b > ΣΩF b .

Let us consider one component B 0 and its preimage A 0 ⊂ A. We construct the homotopy push-out D 0 of the diagram C ← A 0 → B 0 (of connected spaces). The space C and the homotopy fiber F 0 = F ib(A 0 → B 0 ) are simply connected by assumption, which implies that all homotopy fibers over D ₀ are connected. The reduction process explained in Section 2 yields thus a homotopy push-out diagram of connected spaces to which Proposition 4.2 applies.

5. Blackers-Massey for squares

We prove now our main theorem for homotopy push-out squares. The strategy resembles that

of Propostion 4.2, but the rough estimate we obtained there is used to improve the inequality by

introducing it in the formula for cofibrations from Section 3.

Theorem 5.1. We consider a homotopy push-out square of the form A ^f //

g 

B

 h

C ^k // D

and set F = F ib(f ) and G = F ib(g). If the total fiber T is connected, then T > ΩF ∗ ΩG.

Proof. We may assume that D is contractible by Proposition 2.1. As in previous proofs, we show first that one may as well assume that B and C are connected. Indeed, if B (respectively C) were not connected, then G (respectively F ) would not be connected either. The inequality to prove would then simply be T > ΣΩF (respectively T > ΣΩG), which we have established in Theorem 4.3.

Let us thus assume that both B and C are connected. In particular there is a single homotopy fiber F . Let H = F ib(C → C/F ). We will analyze T as the homotopy fiber of the composite F → H → C.

We consider first the homotopy fiber F ib(h : F → H) as the total fiber of the homotopy push- out square given by the cofibration sequence F → C → C/F . Therefore, since T is connected, Proposition 3.1 tells us that F ib(h) > ΩT ∗ ΩT ∨ Ω(T ∗ ΩC). Since C > F and introducing both rough estimates from Theorem 4.3 we obtain

F ib(h) > ΩΣΩF ∗ ΩΣΩG ∨ Ω(ΣΩG ∗ ΩF ) > ΩF ∗ ΩG

where we used the fact that ΩΣX > X for any space X (yet another application of Lemma 1.3).

Let us now move to the second homotopy fiber, namely F ib(H → C) = Ω(C/F ). We have seen in the previous section, at the very end of the proof of Proposition 4.2, that it is killed by Ω(F ∗ ΩB) > Ω(F ∗ ΩG) > ΩF ∗ ΩG.

The total fiber T of the square fits therefore in a fibration sequence where both the base space and the fiber are killed by ΩF ∗ ΩG. Thus so is T .

When we only pay attention to the connectivity of the fibers we obtain as a straightforward corollary the classical triad theorem of Blakers-Massey, [1], or rather its reformulation by Goodwillie in [5, Theorem 2.3].

Theorem 5.2. We consider a homotopy push-out square of the form A ^f //

g



B

 h

C ^k // D

and set F = F ib(f ) and G = F ib(g). If F is n-connected and G is m-connected, then the total

fiber T of the square is (m + n)-connected if it is at least connected.

Proof. The connectivity statement can be reformulated by saying that F > S ⁿ⁺¹ and G > S ^m+1 . The claim follows from the fact that ΩS ⁿ⁺¹ ∗ ΩS ^m+1 > S ^n+m+1 .

Remark 5.3. Example 1.4 shows that the previous theorem is not true in full generality. One has to make some assumption, the best one being the connectivity of T . If one prefers to require only something about the homotopy fibers F and G, one could simply ask them to be connected. This would ensure T to be connected.

References

1. A. L. Blakers and W. S. Massey, The homotopy groups of a triad. II, Ann. of Math. (2) 55 (1952), 192–201.

2. W. Chach´ olski, Desuspending and delooping cellular inequalities, Invent. Math. 129 (1997), no. 1, 37–62.

3. , A generalization of the triad theorem of Blakers-Massey, Topology 36 (1997), no. 6, 1381–1400.

4. T. Ganea, A generalization of the homology and homotopy suspension, Comment. Math. Helv. 39 (1965), 295–322.

5. T. G. Goodwillie, Calculus. II. Analytic functors, K-Theory 5 (1991/92), no. 4, 295–332.

6. M. Mather, Pull-backs in homotopy theory, Canad. J. Math. 28 (1976), no. 2, 225–263.

7. V. Puppe, A remark on “homotopy fibrations”, Manuscripta Math. 12 (1974), 113–120.

8. G. W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New

York, 1978.