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


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1 A NEW ELLULAR VERSION OF LAKERSMASSEY WOJIEH HAHÓLSKI AND JÉRÔME SHERER Abstract. onsider a pushout diaram of spaces A, construct the homotopy pushout, and then the homotopy pullback of the diaram one ets by forettin the initial object A. Wwe compare the difference between A and this homotopy pullback. This difference is measured in terms of the homotopy fibers of the oriinal maps. Restrictin our attention to the connectivity of these maps, we recover the classical lakersmassey Theorem. Introduction This is a very preliminary version. It contains complete proofs, but not all the backround and applications! We write F ib(f) for the homotopy fiber of the map f : A. When is not connected, the homotopy type of these fibers miht vary, dependin on the chosen base point in. In the followin statement F ib(f) means then the wede 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 belons to the smallest collection of spaces closed under weak equivalences, homotopy colimits, and extensions by fibrations. The coaumented functor Id P A is ousfield localization with respect to the map A. Theorem 5.1. We consider a homotopy pushout square of the form A f k D h and set F = F ib(f) and G = F ib(). If the total fiber T of the square is connected, then T > ΩF ΩG. If P denotes the homotopy pullback of the diaram D, 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 pushout square to be a homotopy pullback square. If we focus only on the connectivity of F and G, we obtain the followin consequence, which is Goodwillie s version of the lakersmassey Theorem for squares, [5] Mathematics Subject lassification. Primary 55S45; Secondary 55R15, 55R70, 55P20, 22F50. 1
2 2 WOJIEH HAHÓLSKI AND JÉRÔME SHERER Theorem 5.1. We consider a homotopy pushout square of the form A f k D h If F = F ib(f) is mconnected and G = F ib() is nconnected, then the total fiber is (m + n) connected. We can also prove hiher dimensional, cubical, analoues of the above theorem, which eneralize Goodwillie s hypercubical statements (but they are not in this note). Our theorem also improves the cellular lakersmassey Theorem obtained by the first author [3, Theorem 1.]. There, the inequality was obtained for the homotopy fiber of the suspension of the map A P. It is in eneral impossible to desuspend such an inequality, [2], but always easy to suspend ours. Acknowledments. 1. Notation and known results Given a map f : A, the homotopy fiber depends on the choice of a base point in. If we fix such a base point 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 lobal properties of maps and diarams. 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 wede of F ib b (f) where b runs over a set of points b, one in each connected component of. Given a commutative square A D there is a natural map from A to the homotopy pullback P of the square of which we removed A, i.e. the diaram D. 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 ) F ib( D). Remark 1.1. We can always assume that the homotopy pushout is a connected space. If it is not the case, let us pick a connected component D α D and consider the preimaes A α A,
3 A NEW ELLULAR VERSION OF LAKERSMASSEY 3 α, and α of D α. The fibers and total fibers of the homotopy pushout diaram A α α α D α are some of the fibers and total fibers of the oriinal diaram. Therefore, if the inequality we are lookin for holds for this diaram, for any α, then it holds also for the oriinal diaram. We write X > A, or X (A) and say that A kills X when the nullification functor P A kills X, i.e. P A X is contractible. The followin lemma is in fact the main definin property of the acyclic classes (A), namely the closure under extensions by fibrations. Lemma 1.2. onsider a fibration sequence F E. Then E > F. Recall also that a nonconnected space kills all spaces because the inequality X > S 0 holds for any space X. Lemma 1.3. In a homotopy pushout square, F ib( D) > F ib(a ). Proof. y the previous remark we can suppose that D is connected. Now, if is not connected, the homotopy fiber F ib( D) is not connected either, but since D is connected, there is at least one component in over which the homotopy fiber is not connected. Therefore the inequality F ib( D) > F ib(a ) holds trivially in this case. Let us thus assume that is connected. When 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. Let us order the components and for any cardinal α call α the union of the first α components. We can assume that π 0 A = π 0 (if not, the claim is obvious) and define D α to be the homotopy pushout of the diaram A α α. For a successor β = α + 1 we see that D β is the homotopy pushout of D α D where D is the homotopy pushout of connected spaces A and A β = A α A and β = α. We have therefore a fibration F ib( D α ) F ib( D β ) F ib(d α D β ). y induction hypothesis F ib( D α ) > F ib(a α α ) and from the connected case we deduce that F ib(d α D β ) > F ib( D ) > F ib(a ). This proves the claim for D β. When β is a limit ordinal, we conclude by a similar arument usin a telescope instead of a pushout. Here is an example which illustrates both the necessity to deal with nonconnected spaces and the importance to consider all homotopy fibers at once.
4 4 WOJIEH HAHÓLSKI AND JÉRÔME SHERER Example 1.4. Let f : S 1 S 1 D 2 S 1 be the disjoint union of the inclusion of the circle as boundary of the disc and the identity. onsider the (homotopy) pushout diaram S 1 S 1 D 2 S 1 S 1 D 2 where the left handside vertical map is the fold map (the identity on both copies of the circle). The homotopy pullback 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. (hachólski, [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 pushout diarams in which the terminal object is a point are easier to handle because the homotopy pullback one needs to form in order to compute the total fiber is simply a product. The aim of this section is to reduce the lakersmassey theorem to this situation. Recall from Remark 1.1 that we can always assume that it is connected. Given a commutative pushout square A f k D h we denote by F the homotopy fiber F ib(f) and by G the homotopy fiber F ib(). We will also need the homotopy fibers over D, namely A = F ib(a hf D), = F ib(h) and = F ib(k). Proposition 2.1. For any homotopy pushout square as above the commutative square of homotopy fibers over a connected space D: A f satisfies the followin properties: It is a homotopy pushout square; F ib(f ) F and F ib( ) G; the total fiber T = F ib(a ) is weakly equivalent to T.
5 A NEW ELLULAR VERSION OF LAKERSMASSEY 5 Proof. The first statement is a consequence of Mather s ube Theorem [6], the second and third follow directly from the fact that horizontal (respectively vertical) homotopy fibers aree in homotopy pullback squares. Example 2.2. The typical contractible homotopy pushout is that of the diaram A A where the maps are the obvious collapse maps. Thus the total fiber of the square A A is the homotopy fiber of the inclusion A A. y Puppe s Theorem [7] this is the join ΩA Ω. It is killed by Ω( ΩA) Ω(A Ω) = F ib(a A) F ib(a ) by Lemma 1.3. Remark 2.3. In the previous example we obtained an estimate on the total fiber of a homotopy pushout square by usin the homotopy fibers of the maps to D, before usin the ones from A. This is not to be expected in eneral as shown by the followin example. Let us choose an interal homoloy 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 attachin a sinle 3cell via the attachin map S 2 S 2 S 2 S 2 S 1 S 2 S 1 S 2 where the first map has deree 2 on the first sphere and deree 1 on the second one, and the inclusion is on the zeroth and first factors of the infinite wede. We consider then the homotopy pushout 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. ofibrations 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 pushout square of the form A X X/A
6 6 WOJIEH HAHÓLSKI AND JÉRÔME SHERER 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 throuh A/F and we construct a commutative diaram of homotopy pushout squares A A/F X ΣF X/A Thus the map F Ω(X/A) factors throuh 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. y Ganea s Theorem [4] F ib(a/f X) F ΩX and we conclude thus that T > ΩF ΩF Ω(F ΩX). orollary 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 miht 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 pullback 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 A very rouh estimate In this section we obtain a first very rouh estimate for the total fiber. y comparin 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 pushout
7 A NEW ELLULAR VERSION OF LAKERSMASSEY 7 square A f and call F = F ib(f) and G = F ib(). If b, let F b = F ib b (f). The total fiber T is the homotopy fiber of the map A, but it can be alternatively described as the homotopy fiber F ib(f ). Lemma 4.1. When and are connected, the homotopy cofiber /F of the composite map F A is 2connected if F is simply connected. Proof. We have a homotopy pushout square A/F /F Therefore we infer from Lemma 1.3 that /F = F ib(/f ) is killed by F ib(a/f ) = Ω F. Since F is simply connected this join is 2connected. Here comes the first part of our very rouh estimate. The rouhness 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 lakersmassey Theorem, the connectivity of the total fiber should be related to the sum of the connectivities of both fibers. Proposition 4.2. Assume that 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 is simply connected by Lemma 1.3. y the previous lemma we also know that /F > S 3. The total fiber T is the homotopy fiber of F and this map factors throuh H = F ib( /F ). Let us first analyze the homotopy fiber of the map h : F H. The canonical map F ΩΣF also factors throuh h and we have therefore a fibration sequence F ib(h) F ib(f ΩΣF ) F ib(h ΩΣF ).
8 8 WOJIEH HAHÓLSKI AND JÉRÔME SHERER 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 pushout square /F ΣF and we know by Proposition 3.1 that it is killed by ΩH ΩH Ω(H Ω(/F )), since the homotopy fiber of /F is H, which is killed by F by Lemma 1.3 and thus connected. The inequalities H > F and /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. y the closure property, Lemma 1.2, we infer that F ib(h) > ΣΩF ΩΣ 2 ΩF > ΣΩF. We finally use a similar trick to et the estimate on T, by considerin the composite F H. We obtain here a fibration sequence F ib(h) T F ib(h ). ut the base space is Ω(/F ) and, once aain by Lemma 1.3, /F is killed by F ib(a/f ) = F Ω > ΣF. Thus Ω(/F ) > ΣΩF and we conclude. We remove now the connectivity assumption on the space. Theorem 4.3. If the total fiber T is connected, then T > ΣΩF, and by symmetry T > ΣΩG. Proof. We will actually show somethin a little bit more precise. Just like in the proof of the previous proposition we can assume that F is simply connected, so that is simply connected as well. Therefore f induces an isomorphism on connected components π 0 A = π 0. hoose a point b in any component and let F b denote the correspondin homotopy fiber F ib b (f) and T b the correspondin total fiber F ib (c,b) (A ). We will prove that T b > ΣΩF b. Let us consider one component 0 and its preimae A 0 A. We construct the homotopy pushout D 0 of the diaram A 0 0 (of connected spaces). The space and the homotopy fiber F 0 = F ib(a 0 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 pushout diaram of connected spaces to which Proposition 4.2 applies. 5. lackersmassey for squares We prove now our main theorem for homotopy pushout squares. The stratey resembles that of Propostion 4.2, but the rouh estimate we obtained there is used to improve the inequality by introducin it in the formula for cofibrations from Section 3.
9 A NEW ELLULAR VERSION OF LAKERSMASSEY 9 Theorem 5.1. We consider a homotopy pushout square of the form A f k and set F = F ib(f) and G = F ib(). 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 and are connected. Indeed, if (respectively ) 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 and are connected. In particular there is a sinle homotopy fiber F. Let H = F ib( /F ). We will analyze T as the homotopy fiber of the composite F H. We consider first the homotopy fiber F ib(h : F H) as the total fiber of the homotopy pushout square iven by the cofibration sequence F /F. Therefore, since T is connected, Proposition 3.1 tells us that F ib(h) > ΩT ΩT Ω(T Ω). Since > F and introducin both rouh 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 ) = Ω(/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 Ω) > 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 straihtforward corollary the classical triad theorem of lakersmassey, [1], or rather its reformulation by Goodwillie in [5, Theorem 2.3]. Theorem 5.2. We consider a homotopy pushout square of the form D h A f k and set F = F ib(f) and G = F ib(). If F is nconnected and G is mconnected, then the total fiber T of the square is (m + n)connected if it is at least connected. D h
10 10 WOJIEH HAHÓLSKI AND JÉRÔME SHERER Proof. The connectivity statement can be reformulated by sayin 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 enerality. One has to make some assumption, the best one bein the connectivity of T. If one prefers to require only somethin 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. lakers and W. S. Massey, The homotopy roups of a triad. II, Ann. of Math. (2) 55 (1952), W. hachólski, Desuspendin and deloopin cellular inequalities, Invent. Math. 129 (1997), no. 1, , A eneralization of the triad theorem of lakersmassey, Topoloy 36 (1997), no. 6, T. Ganea, A eneralization of the homoloy and homotopy suspension, omment. Math. Helv. 39 (1965), T. G. Goodwillie, alculus. II. Analytic functors, KTheory 5 (1991/92), no. 4, M. Mather, Pullbacks in homotopy theory, anad. J. Math. 28 (1976), no. 2, V. Puppe, A remark on homotopy fibrations, Manuscripta Math. 12 (1974), G. W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, SprinerVerla, New York, 1978.
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