GENERALIZED FREUDENTHAL SUSPENSION THEOREM AND DE-SUSPENDING AND DE-LOOPING CELLULAR INEQUALITIES

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1 GENERALIZED FREUDENTHAL SUSPENSION THEOREM AND DE-SUSPENDING AND DE-LOOPING 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 [1] and [2]) and E.Dror Farjoun (see [6], [7] and [8]), and has been developed by the latter in his recent book [8]. The key roles are played by two geometric relations between spaces. Here are two surprising facts, which are reflections of these relations: The following theorem is surprising because it says that if A is a suspension, then for any map A X, the homotopy fiber of the cofiber Fib(X X/A) anda have unstably quite a lot in common (see corollary 8.9): Theorem. Let A X X/A be a cofibration sequence of pointed and connected spaces and Y be a pointed space. If A is weakly equivalent to a suspension of a connected space, then the following statements about Y are equivalent: map (A, Y ) is weakly contractible. map (ΩΣA, ( Y ) is weakly ) contractible. map Fib(X X/A),Y is weakly contractible. The following theorem is surprising because of its generality (see corollary 10.3): Theorem. Let F E A be any fibration over a connected space A and h be any homology theory. If F E is an h isomorphism, then A is h acyclic. We are going to study two geometric relations between spaces. These relations are associated to two ways of building one spaces from another: Let us choose a connected space A. By C(A) we denote the smallest class of spaces which contains A, is closed under taking weak equivalences, homotopy push-outs, and arbitrary wedges. We say that X is A-cellularorthatitisbuiltfromA, ifx belongs to C(A). We denote this relation by X A. By establishing the relation X A we will know that X can be built from A using homotopy push-outs and arbitrary wedges. This relation though, does not say how to do it. 1

2 2 WOJCIECH CHACHÓLSKI By C(A) we denote the smallest class of spaces which contains A and is closed under some simple operations. In addition to taking weak equivalences, homotopy push-outs, and arbitrary wedges, as it is in the case of C(A), we require that C(A) is also closed under taking extensions by fibrations. That is, for any fibration sequence F E B, iff C(A) andb C(A), then E C(A). We say that X is A-acyclic or that X is killed by A, ifx belongs to C(A). We denote this relation by X>A. By establishing the relation X>Awe will know that X can be built from A using homotopy push-outs, arbitrary wedges, and extensions by fibrations. As before, this relation does not say how to do it. 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. It is A.K.Bousfield s periodization functor P A (see [1, section 2.8]) and E.Dror Farjoun s colocalization functor CW A (see [6, section 3]). AspaceX is A-acyclic if and only if P A (X) is weakly contractible. Thus the class C(A) coincides with the kernel of the functor P A (see [4, theorem 17.3]). Studying classes C(A) is the same as studying unstable Bousfield classes (see [1, section 9]). AspaceX 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 [4, theorem 8.2]). More surprisingly there is also a functor F : Spaces Spaces, such that X is A-cellular if and only if F (X) is weakly contractible. Therefore the class of A-cellular spaces coincides with the kernel of some functor (see [4, theorem 20.5]). This characterization of classes C(A) andc(a) gives universal properties that describe A-cellular and A-acyclic spaces (see theorem 3.2 and theorem 4.2). 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 p- torsion, 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. We regard two spaces as being close to each other if they satisfy the chosen property. It turns out that many traditional theorems are just reflections of some cellular approximations. Here are some examples of non trivial cellular inequalities: The following is the only general statement, known to me, that relates the homotopy fibers of maps in a homotopy push-out diagram (see [5, theorem 9.4]): Let the following be a homotopy push-out square of connected spaces: A B C D If Fib(A B) is connected, then: Fib(C D) Fib(A B).

3 DE-SUSPENDING AND DE-LOOPING CELLULAR INEQUALITIES 3 Generalized Blakers Massey theorem (see [3, theorem 7.1]): Let the following be a homotopy push-out square of connected spaces: A B Let P = holim(c D B). There is a natural map A P. If Fib(A B) and Fib(A C) are connected, then: Fib(ΣA ΣP ) Fib(A B) Fib(A C). C D Generalized Freudenthal suspension theorem (see theorem 7.2): Let X be connected. If X ΩΣX is the adjoint map to the identity ΣX id ΣX, then Fib(X ΩΣX) ΩX ΩX. Let f :ΣX Y and g : X ΩY be a pair of adjoint maps. If Y is simply connected, then: Fib(f :ΣX Y ) > ΣFib(g : X ΩY ) ΩY ΩY. If Fib(f :ΣX Y ) is simply connected, then: Fib(g : X ΩY ) > ΩFib(f :ΣX Y ) ΩX ΩX. (see corollary 7.3). The goal is to establish various rules for operating on the symbols X A and X> A. Then use these roles in particular cases to show non trivial cellular inequalities. In this paper we are focusing on a question: when can we de-suspend and de-loop cellular inequalities? We say that A allows de-suspending (weak) cellular inequalities if ΣA (>)ΣX implies A (>)X. We say that a simply connected space A allows de-looping (weak) cellular inequalities if ΩX (>)ΩA implies X (>)A. Weprove various characterizations of these properties. In particular we show: A allows de-suspending (weak) cellular inequalities if the class C(A) ( C(A) ) is generated by a loop space (see theorem 8.3) A allows de-looping (weak) cellular inequalities if the class C(A) ( C(A) ) is generated by a suspension of a connected space (see theorem 9.6 and proposition 10.5). We also prove a result, that was originally observed by E.Dror Farjoun. It says that if A allows de-looping weak cellular inequalities, then it allows de-suspending weak cellular inequalities (see corollary 10.6). Thus for any X and Y,Σ n+1 X>Σ n Y is equivalent to ΣX >Y. In preparation for proving these statements, we observe that certain duality is taking place, where we consider A (>) as the dual of X (>)A. The following is an example of almost dual statements that are both true: Theorem. If A X X/A is a cofibration sequence of connected spaces, then: ΩΣA Fib(X X/A) A (see proposition 8.1).

4 4 WOJCIECH CHACHÓLSKI If F E A is a fibration over a connected space A, then E/F ΣΩA, and after only a single suspension: ΣA Σ(E/F) (see theorem 10.1). 2. Notation This paper is written simplicialy. By a space we mean a simplicial set. All spaces except mapping spaces and spaces that are formed as homotopy limits are not assumed to be connected. Thus if we consider a space ΩX or the homotopy fiber F of amapx Y, we do not assume that they are connected, whereas we do assume that X and Y are. Only two exceptions are made: see theorem 3.5 and theorem 9.1 The homotopy cofiber of a map A X is denoted either by X/A, ifitisclear 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 Fib(A X). If X is connected, then the homotopy type of Fib(A X) does not depend on the choice of a basepoint in X. For a space X, byσx we denote the unreduced suspension of X. Ifwechoosea basepoint in X, byωx we denote the homotopy fiber of the basepoint map X. Let X and Y be pointed spaces. The subspace X {y 0 } {x 0 } Y of X Y is denoted by X Y andiscalledthewedgeofx 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 any choice of basepoints in X and Y,thejoinX Y is weakly equivalent to Σ(X Y ). 3. Strong cellular inequalities In this section the definition and properties of the notion of cellular inequalities will be presented. Cellular inequalities were introduced by E. Dror Farjoun (see [7] and [8]). For a detail discussion of this notion see also [3] and [4]. 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 X i C(A). If X i C(A), then hocolim(x 0 X 1 ) C(A). One can think about the class C(A) as generated by A using some simple operations: taking homotopy push-outs and arbitrary wedges (the closure of C(A) under taking telescopes follows from its closure under taking homotopy push-outs and arbitrary wedges).

5 DE-SUSPENDING AND DE-LOOPING CELLULAR INEQUALITIES 5 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 to a strong cellular inequality. If X A, then we will say that X is built from A, orthatx is A-cellular. The relation X A says that there exists a recipe which describes how to built X from A using homotopy push-outs and arbitrary wedges. It does not give any explicit procedure to built X from A. By establishing the relation X A we will learn that for any homotopy property ξ of spaces, which is preserved by homotopy push-outs and arbitrary wedges, if A satisfies ξ, thensodoesx. 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 3.2 (E. Dror Farjoun [6, 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). Remark 3.3. This 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 [4, section 7] and [6, section 3.4]). Thus this universal property identifies the class C(A) as the image of some idempotent functor. The goal is to establish various rules for operating on the symbol X A, and then use those rules in particular cases to show non trivial cellular inequalities. The following proposition and theorem list some basic, although most of them non obvious, examples and rules for operating on the symbol X A. Theorem 3.4. (1) If X is a retract of A, then X A (see [6, section 2.3]). Thus for any A, A. (2) Let the following be a homotopy push-out square: A B If Fib(A B) is connected, then: Fib(C D) Fib(A B). (see [5, theorem 9.4]). C D (3) X ΣA is equivalent to X being simply connected and ΩX A (see [4, theorem 10.8]). Thus ΩΣA A. (4) If X A and Y B, then X Y A B (see [3, corollary 3.10]). In particular since Y S 1 is always true, we have: X Y ΣX.

6 6 WOJCIECH CHACHÓLSKI (5) We can suspend cellular inequalities. If X A, then ΣX ΣA (see [4, corollary 10.2]). (6) We can loop cellular inequalities. If A is simply connected and X A, then ΩX ΩA (see [4, corollary 10.4]). (7) Let f : X Y be a map. Then: Fib(Σf :ΣX ΣY ) Y/X ΣFib(f : X Y ) (see [4, theorem 10.9]). Notice that we are not assuming that Fib(f) is connected. By considering the map A, we obtain that any A is ΣΩA-cellular. In the following theorem we are making an exception from our rule and we are not assuming that involved spaces are connected. Theorem 3.5 ([3, theorem 5.1]). Let B be a connected space. Let us consider the following commutative diagram: H @R l Y E E/Y f p g G X 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. If hocolim(g H F ) is connected, then Z hocolim(g H F ). 4. Weak cellular inequalities In this section the definition and basic properties of the notion of weak cellular inequalities will be presented. This notion was introduced by E.Dror Farjoun (see [7] and [8]). For a detail discussion of this notion see also [4]. Definition 4.1. 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).

7 DE-SUSPENDING AND DE-LOOPING CELLULAR INEQUALITIES 7 If X i C(A), then hoclim(x 0 X 1 X 2 ) C(A). If X i C(A), then X i C(A). If X i C(A), then hocolim(x 0 X 1 ) C(A). If F E B is a fibration sequence and F, B C(A), then E C(A). One can think about the class 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. Notation. If X C(A), then we will write X>A, and call the relation > a weak cellular inequality. If X>A,thenwesaythatX is killed by A or that X is A-acyclic. The relation X>Asays that we can built X from A using homotopy push-outs, arbitrary wedges and extensions by fibrations. It does not say how to do it. By establishing the relation X>Awe will learn that for any homotopy property ξ of spaces, which is preserved by homotopy push-outs, arbitrary wedges, and extensions by fibrations, if A satisfies ξ, then so does X. 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 4.2 ([4, 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,ifthe mapping space map (A, Y ) is weakly contractible, then so is map (X, Y ). Remark 4.3. This 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 [1, section 2.8]). Thus this universal property identifies the class C(A) as the kernel of some idempotent functor (see [4, theorem 17.3]). We will use the following properties of the relation >: Theorem 4.4. (1) X>ΣAis equivalent to X being simply connected and ΩX >A(see [4, theorem 18.5] and compare with theorem 3.4.(3)). (2) Let F E B be a fibration sequence. If F A and B>ΣA, then E A (see [5, corollary 8.4]). It is obvious from the definitions that if X A, thenx > 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, thus when n =1, 3, 7 (see [4, corollary 20.13]).

8 8 WOJCIECH CHACHÓLSKI 5. Ganea s theorem In this section a generalized Ganea theorem is going to be presented. Although this theorem has been known, we will present its proof. We start with stating two well known properties of homotopy fibers. For the reference see [7]. Proposition 5.1. If the following be a commutative diagram, then: X Y Fib ( hocolim(z X Y ) W ) hocolim ( Fib(Z W ) Fib(X W ) Fib(Y W ) ). Z W Let (X 0 X 1 X 2 ) be a telescope diagram. Then: Fib ( X 0 hocolim(x 0 X 1 ) ) hocolim ( Fib(X 0 X 1 ) Fib(X 0 X 2 ) ) As a corollary we get Ganea s theorem: Theorem 5.2 (Ganea). Let X 0 and X 1 be spaces. There is a weak equivalence: Fib(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 introduced 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 : X l X l, (x 0,...,x n 1 ) (x 0,...x k 1,,x k,...,x n 1 ) l k 0 l n 0 l n Let W be the union of all the images of i k.thusw is the 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 5.3. Fib(W 0 l n X l ) ΩX 0 ΩX 1 ΩX n. Proof. The proof is by induction on n. Forn = 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 ( ) X l V V X n 0 l n 1

9 DE-SUSPENDING AND DE-LOOPING CELLULAR INEQUALITIES 9 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 5.1, we get: Fib ( W ( pr hocolim ΩX 1 n ΩXn Fib ( V 0 l n 0 l n 1 X l ) ) pr2 ( X l Fib V ) ) X l 0 l n 1 By inductive assumption Fib(V 0 l n 1 X l ) ΩX 0 ΩX 1 ΩX n 1.Therefore: Fib(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 a 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 the map X hocolim(j 1 J 2 )isweakly equivalent to the map X ΩΣX, which is adjoint to the identity ΣX id ΣX (see [9, theorem 2.6]). (n+1) def Theorem 6.1. Fib(J n J n+1 ) (ΩX) =ΩX } ΩX{{ ΩX}. n+1 Proof. According to generalized Ganea s theorem (see 5.3): Using theorem 3.4.(2) we can conclude: Fib(W (n) X n+1 ) (ΩX) (n+1) Fib(J n J n+1 ) Fib(W (n) X n+1 ) (ΩX) (n+1)

10 10 WOJCIECH CHACHÓLSKI 7. Generalized Freudenthal Suspension Theorem 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: ΩX ΩX X ΣΩX the natural map ΣΩX hocolim( ΩX ) X is adjoint to the identity map ΩX id ΩX. Therefore by proposition 5.1: Corollary 7.1. Fib(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: X ΩΣX ΣX ΣX 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 Generalized Freudenthal Suspension Theorem: Theorem 7.2. Fib(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 Fib(X J n ) ΩX ΩX. SinceJ 1 = X and the map X J 1 is the identity, the statement is true for n =1. Let us assume that it is true for n 1. We will show that Fib(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: Fib(X J n ) Fib(X J n+1 ) Fib(J n J n+1 ) By inductive assumption Fib(X J n ) ΩX ΩX. According to theorem 6.1: Fib(J n J n+1 ) (ΩX) (n+1)

11 DE-SUSPENDING AND DE-LOOPING CELLULAR INEQUALITIES 11 and thus by theorem 3.4.(4): Fib(J n J n+1 ) ΣΩX ΩX. It follows from theorem 4.4.(2), thatfib(x J n+1 ) ΩX ΩX. To conclude the proof, observe (see proposition 5.1): Fib(X ΩΣX) Fib ( X hocolim(j 1 J 2 ) ) hocolim ( Fib(X J 1 ) Fib(X J 2 ) ) ΩX ΩX Corollary 7.3. Let f :ΣX Y, g : X ΩY be a pair of adjoint maps. If Y is simply connected, then: Fib(f :ΣX Y ) > ΣFib(g : X ΩY ) ΩY ΩY If Fib(f :ΣX Y ) is simply connected, then: Fib(g : X ΩY ) > ΩFib(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 than compose it with the natural map ΣΩY Y : (ΣX It follows that we have a fibration sequence: Fib(ΣX Σg ΣΩY ) Fib(ΣX f Y )=(ΣX Σg ΣΩY Y ) f Y ) Fib(ΣΩY Y ) Simply connectedness of Y gives connectedness of ΩY, and thus theorem 3.4.(7) implies: Fib(Σg :ΣX ΣΩY ) ΣFib(g : X ΩY ). Since Fib(ΣΩY Y ) ΩY ΩY, it follows that Fib(f :ΣX Y ) > ΣFib(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 natural map X ΩΣX: (X g ΩY )=(X ΩΣX Ωf ΩY ) Therefore we have a fibration sequence with a connected base: Fib(X ΩΣX) Fib(X g ΩY ) Fib(ΩΣX Ωf ΩY ) Since Fib(X ΩΣX) ΩX ΩX (see theorem 7.2), we get: Fib(g : X ΩY ) > ΩFib(f :ΣX Y ) ΩX ΩX

12 12 WOJCIECH CHACHÓLSKI 8. De-suspending cellular inequalities In this section we are going to discuss under what circumstances we can de-suspend cellular inequalities. We start with a preliminary result: Proposition 8.1. Let A X X/A be a cofibration sequence. Then: ΩΣA Fib(X X/A) A Proof. Consider the following two homotopy push-out squares: A X X/A X X/A ΣA By applying theorem 3.4.(2) we can conclude: ΩΣA Fib(X X/A) A. Definition 8.2. We say that a space A allows de-suspending (weak) cellular inequalities if ΣA (>)ΣX implies A (>)X. The following theorem shows that the problem of de-suspending cellular inequalities is closely related to reversing the inequalities stated in proposition 8.1: Theorem 8.3. The following statements are equivalent: (1) A allows de-suspending (weak) cellular inequalities. (2) A (>)ΩΣA. (3) C(A) =C(ΩΣA) ( C(A) =C(ΩΣA) ). (4) If A X X/A is a cofibration sequence, then A (>)Fib(X X/A). (5) If A X X/A is a cofibration sequence, then C(A) =C ( Fib(X X/A) ) ( ) C(A) =C(Fib(X X/A)). (6) There exists a simply connected space B, such that C(A) =C(ΩB) ( C(A) = C(ΩB) ). Proof. (1) (2). Notice that by theorem 3.4.(3) the inequality ΩΣA ΩΣA is equivalent to ΣA ΣΩΣA. Thus since A allows de-suspending (weak) cellular inequalities, we get: A (>)ΩΣA. (2) (3) (4) (5) (6). Those implications are either obvious or are easy corollaries of proposition 8.1. (6) (1). By looping the inequality B ΣΩB (see theorem 3.4(7)), we get: ΩB ΩΣΩB. Let us assume that ΣA (>)ΣX. It follows from theorem 3.4.(3) (theorem 4.4.(1))that ΩΣA (>)X, and therefore A (>)ΩB ΩΣΩB (>)ΩΣA (>)X.

13 DE-SUSPENDING AND DE-LOOPING CELLULAR INEQUALITIES 13 Remark 8.4. Let us consider the following property of a space A: If A X X/A is a cofibration sequence, then C(A) =C ( Fib(X X/A) ) ( ) C(A) =C(Fib(X X/A)). Theorem 8.3 indicates that this property depends only on the class C(A) ( C(A) ) generated by A, rather than on the homotopy type of A (compare with remark 9.7 and remark 10.8). Theorem 8.3 gives us a simple procedure of constructing new spaces that allow de-suspending (weak) cellular inequalities out of old ones: Corollary 8.5. For a simply connected spaces B, ΩB allows de-suspending cellular inequalities. Let A X X/A be a cofibration sequence. If we can de-suspend (weak) cellular inequalities with A, then so can we with Fib(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 de-suspend cellular inequalities. If A is an H-space, it is a retract of ΩΣA, and thus we have a strong cellular inequality A ΩΣA (see theorem 3.4.(1)). Therefore H-spaces allow de-suspending cellular inequalities. In general though, it is difficult to expect that there is a strong cellular inequality A ΩΣA, since according to [4, corollary 20.13] S n ΩS n+1 if and only if S n is an H-space. However, it is alwaystruethatwehaveaweak cellular inequality S n > ΩS n+1. It was originally observed by E.Dror Farjoun, that in general suspensions do allow de-suspending weak cellular inequalities: Theorem 8.6. If C(A) =C(ΣB), then A allows de-suspending weak cellular inequalities. Proof. Since the property of allowing de-suspending weak cellular inequalities depends only on the class C(A), we can assume that A ΣB. To prove the theorem, we have to show that A>ΩΣA. Therefore it would be enough to prove that the homotopy fiber Fib(A ΩΣA) is built from ΩΣA. Using Generalized Freudenthal Suspension Theorem (see theorem 7.2) we will show more: Lemma 8.7. If A ΣB, then Fib(A ΩΣA) ΣΩΣA. Proof. By theorem 7.2 Fib(A ΩΣA) ΩA ΩA. We will prove the lemma, if we show that ΩA ΩA ΣΩΣA. By suspending inequality ΩΣB B, weget: ΣΩA ΣΩΣB ΣB A. Since ΩA ΩA ΣΩA ΩA, theorem 3.4.(4) implies ΩA ΩA ΣΣΩA, and therefore, by the previous argument ΩA ΩA ΣΩΣA. Notice that ΣA ΣΩΣA, and thus the lemma is proven. Corollary 8.8. Σ n+1 A>Σ n B if and only if ΣA >B.

14 14 WOJCIECH CHACHÓLSKI Corollary 8.9. Let A X X/A be a cofibration sequence, Y be a Kan simplicial set and ξ be any property of spaces that is preserved under taking homotopy pushouts, arbitrary wedges and extensions by fibrations. If A ΣB, then A allows desuspending weak cellular inequalities. Therefore: C(A) =C(Fib(X X/A)) = C(ΩΣA) It follows that A satisfies ξ if and only if Fib(X X/A) does. In particular the following statements about Y are equivalent: map (A, ( Y ) is weakly contractible. ) map Fib(X X/A),Y is weakly contractible. map (ΩΣA, Y ) is weakly contractible. In the next two sections the question of de-looping cellular inequalities will be addressed. We will discus separately the cases of strong and weak cellular inequalities. 9. De-looping strong cellular inequalities We start with a theorem, which says that under some rather general assumptions the dual statement to theorem 3.4.(2) is true, where X A is consider to be the dual of A X. In the following theorem we are making an exception from our rule and we are not assuming that involved spaces are connected. Theorem 9.1. Let the following be a homotopy pull-back square: Y E p X B If p is a principal fibration and E/Y is connected, 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: X B K id K Notice that Fib(B K) E, Fib(X K) Y and Fib(K id K). Therefore, according to theorem 3.5: B/X Fib(B/X K/K) hocolim( Y E) E/Y. Corollary 9.2. Let F E A be a fibration. Then E/F ΣΩA.

15 DE-SUSPENDING AND DE-LOOPING CELLULAR INEQUALITIES 15 Proof. Let us consider the following homotopy pull-back square: ΩA F E Since the map E is a principal fibration and /ΩA ΣΩA is connected, we can applying theorem 3.5, obtaining: E/F ΣΩA. Corollary 9.3. If F E A is a principal fibration, then A E/F ΣΩA. Proof. The second inequality is a consequence of corollary 9.2. To show the first inequality consider the following homotopy pull-back square, and apply theorem 9.1: F E A Remark 9.4. 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 fact, in the next section we will prove a general, slightly weaker statement, that for any fibration F E A after only a single suspension we have: ΣA Σ(E/F) (see theorem 10.1). Definition 9.5. We say that a simply connected space A allows de-looping strong cellular inequalities if ΩX ΩA implies X A. The following theorem shows that the problem of de-looping cellular inequalities is closely related to reversing the inequalities stated in corollary 9.3: Theorem 9.6. Let A be simply connected. The following statements are equivalent: (1) A allows de-looping strong cellular inequalities. (2) ΣΩA A. (3) C(A) =C(ΣΩA). (4) If F E A is a fibration, then E/F A. (5) If F E A is a principal fibration, then C(A) =C(E/F). (6) There exists a space B, suchthatc(a) =C(ΣB).

16 16 WOJCIECH CHACHÓLSKI Proof. (1) (2). By theorem 3.4.(3) inequality ΣΩA ΣΩA is equivalent to ΩΣΩA ΩA. Therefore since A allows de-looping cellular inequalities, we get: ΣΩA A. (2) (3) (4) (5) (6). Those implications are either obvious or are easy consequences of corollary 9.2 and corollary 9.3. (6) (1). By suspending the inequality ΩΣB B, we get: ΣΩΣB ΣB. Let us assume that ΩX ΩA. It follows from theorem 3.4.(3) that X ΣΩA and therefore X ΣΩA ΣΩΣB ΣB A. Remark 9.7. Let us consider the following two properties of a space A: (1) If F E A is a principal fibration, then C(E/F)=C(A). (2) If F E A is a fibration, not necessarily principal, then C(E/F)=C(A). Theorem 9.6 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. SinceA is a suspension, by theorem 9.6, ΣΩA A. Therefore suspensions satisfy property (2). If this property depends only on the class C(A), then in the theorem 9.6 we would not have to restrict ourselves to principal fibrations. 10. De-looping weak cellular inequalities We start with proving that after only a single suspension the dual statement to proposition 8.1 is also true: Theorem Let F E p A be a fibration, possibly non principal. Then ΣA Σ(E/F). Lemma Let f : X X Y be the graph of the map f : X Y, i.e. f(x) = (x, f(x)). For any choice of a basepoint in Y, Σ ( (X Y )/ f(x) ) Σ(X Y ). Therefore the suspension of the homotopy cofiber of the graph f depends only on the spaces X and Y, and does not depend on the map f. Proof. Let us choose a basepoint in Y,andlet : X Y be the trivial map. Consider the following two sets of composable maps: X X Y p 1 X, X f X Y p 1 X Notice that the composition of these both sets of maps gives the identity, therefore: Σ(X Y ) Σ ( (X Y )/ (X) ) Cof(X Y p 1 X) Σ ( (X Y )/ f(x) )

17 DE-SUSPENDING AND DE-LOOPING CELLULAR INEQUALITIES 17 Proof of the theorem. Consider the following commutative diagram: p E E A p p 2 According to theorem 3.5: A id (E A)/ p(e) Fib ( (E A)/ p(e) A/A ) hocolim( F E) E/F By suspending this inequality and applying the lemma, we get: Σ(E A) Σ ( E A)/ p(e) ) Σ(E/F) Since A is a retract of E A, wehave:a E A, and thus ΣA Σ(E/F). Corollary Let F E A be a fibration and h be any homology theory. If F E is h isomorphism, then A is h -acyclic (notice that we are not making any assumptions about the fibration). Definition We say that a simply connected space A allows de-looping weak cellular inequalities if ΩX > ΩA implies X > A. Proposition Let A be simply connected. The following statements are equivalent: (1) A allows de-looping weak cellular inequalities. (2) ΣΩA >A. (3) C(A) =C(ΣΩA). (4) There exists a space B, suchthatc(a) =C(ΣB). Proof. (1) (2). By theorem 3.4.(3) we have: ΩΣΩA ΩA. SinceA allows de-looping weak cellular inequalities,s ΣΩA >A. (2) (3) (4). Those implications are either obvious or are easy corollaries of theorem 3.4.(7). (4) (1). LetΩX>ΩA. By theorem 4.4.(1), weget: X>ΣΩA >ΣΩΣB >ΣB > A. Corollary 8.9 together with the above proposition imply: Corollary Let A be simply connected. If A allows de-looping weak cellular inequalities, then it allows de-suspending weak cellular inequalities. Theorem Let A be simply connected. The following statements are equivalent: (1) A allows de-looping weak cellular inequalities. A

18 18 WOJCIECH CHACHÓLSKI (2) If F E A is a fibration sequence, then E/F > A. (3) If F E A is a fibration sequence, then C(A) =C(E/F). Proof. (1) (2). SinceA allows de-looping weak cellular inequalities, therefore ΣΩA > A (see proposition 10.5). Corollary 9.2 implies: E/F ΣΩA >A. (2) (3). WehavetoshowthatA>E/F. By theorem 10.1 ΣA Σ(E/F). Since A allows de-looping weak cellular inequalities, then it also allows de-suspending weak cellular inequalities (see corollary 10.6), therefore A>E/F. (3) (1). Consider the loop fibration ΩA A. By assumption C(A) = C(ΣΩA), and thus by proposition 10.5 A allows de-looping weak cellular inequalities. Remark Let us consider the following property of a simply connected space A: If F E A is a fibration, then C(A) =C(E/F). Theorem 10.7 indicates that this property depends only on the class C(A) generated by A, rather than on the homotopy type of A (compare with remarks 8.4 and 9.7). References 1. A.K.Bousfield, Localization and periodicity in unstable homotopy theory, preprint. 2. A.K.Bousfield, Unstable localization and periodicity, Proc. to the conf. in Alg. Top. Barcelona, Summer W. Chachólski, A generalization of the triad theorem of Blakers-Massey, Submitted for publication, W. Chachólski, On the functors CW A and P A, to appear in Duke Journal of Mathematics. 5. W.Chachólski, Closed classes, Proc. to the conf. in Alg. Top. Barcelona, Summer E.Dror Farjoun, Cellular spaces, preprint. 7. E.Dror Farjoun, Cellular inequalities, Proc. to the conf. in Alg. Top. Northeastern Univ. June 1993, Springer Verlag. 8. E.Dror Farjoun, Cellular Spaces, Null Spaces and Homotopy Localization, Lecture Notes in Mathematics 1622, Springer Verlag G. W. Whitehead, Elements of Homotopy Theory, Grad. Texts in Math. 61, Springer Verlag The Fields Institute, 222 College St., University of Toronto, Toronto, Ontario M5T 3J1, Canada address: wchachol@fields.toronto.edu