FUNCTORIAL DECOMPOSITIONS OF LOOPED COASSOCIATIVE CO-H SPACES. 1. Introduction

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1 FUNCTORIAL DECOMPOSITIONS OF LOOPED COASSOCIATIVE CO-H SPACES PAUL SELICK, STEPHEN THERIAULT, AND JIE WU Abstract. Selick ad Wu gave a fuctorial decompositio of ΩΣX for pathcoected, p-local CW -complexes X which obtaied the smallest otrivial atural retract A mi (X) of ΩΣX. This paper uses methods developed by the secod author i order to exted such fuctorial decompositios to the loops o coassociative co-h spaces. 1. Itroductio Let X be a path-coected, p-local CW -complex. Selick ad Wu [SW1, SW2] gave a fuctorial decompositio ΩΣX A mi (X) ΩB max (X), where A mi (X) is the smallest atural retract whose homology cotais the homology of X. This homotopy decompositio is the geometric realizatio of a more geeral algebraic result, which obtais the smallest atural coalgebra retract of a tesor algebra. The trasitio from algebra to geometry is suggested by the Bott-Samelso Theorem, which gives a algebra isomorphism H (ΩΣX) = T ( H (X)) (homology with mod-p coefficiets). The Bott-Samelso Theorem ca be geeralized to co-h spaces. If Y is a simply coected co-h space the there is a algebra isomorphism H (ΩY ) = T (Σ 1 H (Y )). The questio arises whether Selick ad Wu s fuctorial decompositio of ΩΣX ca be geeralized to the case of ΩY. A questio i the same spirit was addressed i [T]. There, the fuctorial decompositio ΣΩΣX =1 ΣX() was geeralized to the case of a coassociative co-h space Y. It was show that ΣΩY =1 M (Y ) for spaces M (Y ) satisfyig Σ 1 M (Y ) Y (). The purpose of this paper is to show that the homotopy decompositio of Selick ad Wu geeralizes to the case of a coassociative co-h space Mathematics Subject Classificatio. Primary 55P35. Keywords ad Phrases. Homotopy decompositio, coassociative co-h spaces 1

2 Theorem 1.1. Let Y be a simply coected, homotopy coassociative co-h space. The there is a space M (Y ), a homotopy fibratio A mi (Y ) ad a homotopy decompositio M (Y ) Y, ΩY A mi (Y ) ΩM (Y ) where H (A mi (Y )) is the smallest atural coalgebra retract of H (ΩY ) = T (Σ 1 H (Y )). This homotopy decompositio is atural for co-h maps Y Z betwee homotopy coassociative co-h spaces. This paper is orgaized as follows. I Sectio 2 we review the work of Selick ad Wu which gives fuctorial coalgebra decompositios of tesor algebras ad loop suspesios. I Sectio 3 we review the costructios i [T]. Selick ad Wu s result depeds o particular wedge decompositios of ΣX (). The wedge summads are obtaied as telescopes of self-maps of ΣX () which arise from the actio of the symmetric group o letters. I Sectio 4 we show that such wedge decompositios ca be geeralized to a decompositio of M (Y ) whe Y is a coassociative co-h space. Fially, i Sectio 5 we prove Theorem Fuctorial coalgebra decompositios of tesor algebras ad loop suspesios The material i this sectio comes from [SW1, SW2]. Let V be a vector space over a field k of characteristic p. Let T (V ) be the tesor algebra geerated by V ; this becomes a Hopf algebra by lettig the elemets of V be primitive. Let L (V ) be the set of homogeeous Lie elemets of tesor legth i T (V ). Theorem 2.1. There are fuctorial submodules (V ) of L (V ) such that, if we defie B max (V ) = T ( (V )), the: =2 (a) B max (V ) is a sub-hopf algebra of T (V ), (b) L (V ) B max (V ) if is ot a power of p, 2

3 (c) there is a atural coalgebra decompositio T (V ) = A mi (V ) B max (V ), where A mi (V ) is the smallest atural coalgebra retract of T (V ). Let B max (V ) = B max T (V ). The modules ad B max are realized via idempotets o V. Let be defied by the iterated commutator β : V V β (a 1 a ) = [a 1, [a 2, [a 1, a ] ]. Let Σ be the symmetric group o letters. Let Z (p) be the p-local itegers. Let Z (p) [Σ ] be the group rig. We ca idetify β with the correspodig elemet i Z (p) [Σ ]. The there are elemets α max, δ max Z (p) [Σ ] such that λ max = β α max ad δ max are idempotets i Z (p) [Σ ], ad (V ) = Im(λ max : V V ) B max (V ) = Im(δ max : V V ). The coalgebra decompositio i Theorem 2.1 ca be realized geometrically. We use homology with mod-p coefficiets throughout. Recall that if X is a path coected space the H (ΩΣX) = T ( H (X)). Theorem 2.2. Let X be a path-coected p-local CW -complex. The there are homotopy fuctors such that the followig properties hold: ad A mi from path-coected p-local CW complexes to spaces (a) (X) is a fuctorial retract of ΣX (), (b) there is a fuctorial fiber sequece A mi (X) =2 (X) π X ΣX, (c) there is a fuctorial homotopy decompositio ΩΣX A mi (X) Ω( 3 =2 (X)),

4 (d) there is a fuctorial coalgebra filtratio o H (A mi (X)) such that there is a fuctorial isomorphism of coalgebras Gr H (A mi (X)) = A mi ( H (X)). To describe how the algebraic decompositio i Theorem 2.1 traslates ito the geometric decompositio i Theorem 2.2, observe that a elemet σ Σ correspods to a map σ : X () X () by permutig the factors i the smash product. Suspedig, such maps ca be added, so the idempotets λ max, δ max Z (p) [Σ ] correspod to maps Let The λ max, δ max : ΣX () ΣX (). (X) = hocolim λ max ΣX () B max (X) = hocolim δ max ΣX (). H ( H (B max (X)) = ( H (X)) (X)) = B max ( H (X)). Similarly, 1 λ max, 1 δ max Z (p) [Σ ] are idempotets. Let Sice λ max (X) = hocolim 1 λ max ΣX () B max (X) = hocolim 1 δ max ΣX (). + (1 λ max ) = 1 while λ max ΣX () (1 λ max ) = 0, we get a sum (X) (X) which is a homology isomorphism ad therefore a homotopy equivalece. Similarly, there is a homotopy equivalece ΣX () B max (X) B max (X). 4

5 Let w : ΣX () ΣX be the -fold iterated Whitehead product of the idetity map with itself. Let (X) ΣX () be a right homotopy iverse for the hocolim map ΣX () (X). Let π,x be the composite Let π,x : (X) ΣX (X) = Let π X be the wedge sum of the π,x s: Note that H (Ω (X)) = H (Ω( =2 () αmax ΣX () =2 (X). π X : (X) ΣX. (X))) = T ( = =2 w ΣX. ( H (X)) B max ( H (X)) = H (B max (X)). Thus, if A mi (X) is defied as the homotopy fiber of π X the the homotopoy decompositio i Theorem 2.2 ad the coalgebra decompositio i Theorem 2.1 combie to show there is a coalgebra isomorphism H (A mi (X)) = A mi ( H (X)). 3. The costructio ad properties of the spaces M (Y ) Let Y be a homotopy coassociative co-h space. This sectio reviews the costructio of the space M (Y ) i [T] ad describes some the properties prove there. We also take the opportuity here to prove two additioal properties of M (Y ) which should really have bee icluded i [T]. These will subsequetly be eeded i Sectio 4. They are Lemmas 3.5 ad 3.8. We first record three geeral facts about co-h spaces (see [G]). Lemma 3.1. The followig hold: (a) A space Y is a co-h space if ad oly if there is a map s : Y ΣΩY which is a right homotopy iverse of the evaluatio map ΣΩY Y. (b) A co-h space Y is homotopy coassociative if ad oly if the map s i part (a) ca be chose to be a co-h map. 5

6 (c) If f : Y Z is a co-h map betwee homotopy coassociative co-h spaces the there is a homotopy commutative diagram Y s Y ΣΩY f Z sz ΣΩZ where s Y ad s Z are both co-h maps. ΣΩf Suppose X is a coected space. Oe cosequece of the James costructio is a homotopy equivalece ΣΩΣX =1 ΣX () which is atural for maps X X. The followig theorem geeralizes this decompositio from suspesios to coassociative co-h spaces. Theorem 3.2. Let Y be a simply coected, homotopy coassociative co-h space. The for each 1 there are spaces M (Y ) ad a homotopy equivalece ΣΩY M (Y ) where: (a) Σ 1 M (Y ) Y (), =1 (b) if Y = ΣY the M (Y ) ΣY (), (c) each M (Y ) is a homotopy coassociative co-h space, (d) if Y is also homotopy cocommutative the so is each M (Y ), (e) this homotopy decompositio is atural for co-h maps Y Z betwee homotopy coassociative co-h spaces. Each M (Y ) is costructed as a telescope of a idempotet γ o Σ(ΩY ) (). The basic composite to work with is θ : ΣΩY ev Y s ΣΩY, where ev is the evaluatio map ad s is the co-h map i Lemma 3.1 (b). Shiftig the suspesio coordiate o Σ(ΩY ) () to the i th smash factor, we ca do θ o the i th smash factor ad the idetity map o the remaiig factors. Do this oce for each 1 i. The composite of all iteratios defies γ : Σ(ΩY ) () Σ(ΩY ) (). Let X = ΩY. Let M (Y ) = hocolim γ ΣX (). Let r : ΣX () M (Y ) be the telescope map. 6

7 Sice γ is a idempotet, r has a right homotopy iverse s : M (Y ) ΣX (). Note that whe = 1 the map s is just the co-h structure map Y Whe > 1 the map s has two properties aalogous to those of s = s 1. Propositio 3.3. The map M (a) s ca be chose to be a co-h map, s ΣX () has the followig properties: s ΣΩY = ΣX. (b) there is a isomorphism H (M (Y )) = Σ(Σ 1 H (Y )) ad (s ) icludes H (M (Y )) ito H (ΣX () ) = Σ H (X) by the -fold tesor iclusio. We ext describe aturality. Suppose f : Y Z is a co-h map betwee homotopy coassociative co-h spaces. We cotiue to use X = ΩY. Let X = ΩZ. Let g : X X be Ωf. Let M (f) be the composite M (f) : M (Y ) s ΣX () Σg () ΣX () r M (Z). Lemma 3.4. The costructio of M ( ) is atural for co-h maps Y Z betwee homotopy coassociative co-h spaces. That is, there are homotopy commutative diagrams M (Y ) s ΣX () ΣX () r M (Y ) M (Z) M (f) s Σg () Σg () ΣX () ΣX () r M (Z) M (f) We ow prove a additioal feature of the M ( ) s. I Lemma 3.5 we will show that M (f) is a co-h map. Note that this is ot immediate from the defiitio of M (f) as the composite r Σg () s. By Propositio 3.3 (a), s is a co-h map, while Σg () is a co-h map because it is a suspesio. But r is defied i part by evaluatio maps ad is ot co-h. So it is ot true that M (f) is co-h because it is the composite of co-h maps. Nevertheless, we have: Lemma 3.5. The map M (Y ) M(f) M (Z) is a co-h map. Proof. Argue exactly as i [T, 7.2] (which shows a certai other map is co-h, i fact, the map M (σ) appearig below i Propositio 3.7 (c)). 7

8 Corollary 3.6. M defies a fuctor from the category of homotopy coassociative co-h spaces ad co-h maps to itself. We ext see how the symmetric group Σ acts o M (Y ). Let σ Σ. There is a self-map σ : X () X () give by permutig the factors i the smash product. Let M (σ) be the composite From [T] we have: M (σ) : M (Y ) s ΣX () Propositio 3.7. The followig hold: Σσ ΣX () (a) there is a homotopy commutative diagram r M (Y ). M (Y ) M(σ) M (Y ) s ΣX () Σσ ΣX (), (b) if σ 1, σ 2 Σ the M (σ 1 σ 2 ) M (σ 1 ) M (σ 2 ), (c) M (σ) is a co-h map, (d) (M (σ)) acts o H (M (Y )) = Σ(Σ 1 H (Y )) by permutig the s tesor factors. We ow show that M (σ) is atural. Agai, suppose f : Y Z is a co-h map betwee homotopy coassociative co-h spaces. Agai, let X = ΩY, X = ΩZ, g = Ωf, ad recall the defiitio of the map M (Y ) M(f) M (Z) (which preceeds Lemma 3.4). Lemma 3.8. There is a homotopy commutative diagram M (Y ) M (σ) M (Y ) M (Z) M (f) M (σ) M (f) M (Z). Proof. To keep track of the space ivolved, deote the maps M (Y ) s ΣX () M (Y ) by s Y ad r Y respectively. Recall that r Y s Y is homotopic to the idetity o M (Y ). As well, deote the map M (Y ) M(σ) M (Y ) by M Y (σ). 8 r

9 Cosider the diagram M Y (σ) M (Y ) s Y s Y M (Y ) ΣX () Σσ M (f) s Z Σg () Σσ ΣX () Σg () M (Z) ΣX () ΣX () r Z M (Z) M Z(σ) s Z M (Z) The top ad bottom triagles homotopy commute by Propositio 3.7 (a). The left ier square homotopy commutes by the aturality i Lemma 3.4 while the right ier square homotopy commutes by the aturality of Σσ. Thus the etire diagram homotopy commutes. By defiitio, M (f) = r Z Σg () s Y so the upper directio aroud the diagram is homotopic to M (f) M Y (σ). Sice r Z s Z is homotopic to the idetity o M (Z), the lower directio aroud the diagram is homotopic to M Z (σ) M (f). Thus M (f) M Y (σ) M Z (σ) M (f), provig the Lemma. Fially, we ed this sectio with more review material from [T]. Oe applicatio of the spaces M (Y ) is to costruct geeralizatios of Whitehead products. For a space X, recall that w : ΣX () ΣX is the -fold iterated Whitehead product of the idetity map with itself. Cosider the special case whe X = ΩY. Let ev : ΣX = ΣΩY Y be the evaluatio map. Let s : Y ΣΩY = ΣX be the co-h map i Lemma 3.1 (b). Defie a geeralized Whitehead product o Y by the composite w : M (Y ) s ΣX () w ΣX ev Y. This geeralized Whitehead product w is compatible with w : 9

10 Lemma 3.9. There is a homotopy commutative diagram M (Y ) w Y s ΣX () w ΣX. s Further, the map w is atural for co-h maps Y Z betwee homotopy coassociative co-h spaces.. 4. Idempotet decompositios of ΣX () ad M (Y ) I this sectio we cosider wedge decompositios of ΣX () which arises from the actio of the symmetric group Σ, ad show aalogous wedge decompositios exists for M (Y ). We begi with a defiitio. Defiitio 4.1. Maps f 1,..., f k Z (p) [Σ ] give a orthogoal decompositio of the idetity if: (1) k i=1 f i = 1, (2) f i f i = f i for 1 i k, ad (3) f i f j = 0 wheever i j. To each σ Σ there correspods a map σ : X () X () give by permutig the factors i the smash product. I order to add such maps we eed to susped. However, while [ΣX (), ΣX () ] is a group it is ot ecessarily commutative (it will be, for example, if X is a suspesio). So i geeral, there is o represetatio Z (p) [Σ ] [ΣX (), ΣX () ]. Nevertheless, give α = a 1 σ a! σ! Z (p) [Σ ], we ca still defie a map α : ΣX () ΣX (). It just has to be remembered that the homotopy class of α depeds o the order of σ 1,..., σ!. Oce we take homology, however, the o-commutativity problem goes away. There is a represetatio Z (p) [Σ ] Hom(H (ΣX () ), H (ΣX () )). I particular, suppose f 1,..., f k Z (p) [Σ ] is a orthogoal decompositio of the idetity. The the maps f i : ΣX () ΣX () 10

11 have the property that (f 1 ),..., (f k ) is a orthogoal decompositio of the idetity i Hom(H (ΣX () ), H (ΣX () )). Let Q,i (X) = hocolim fi ΣX (). The H (Q,i (X)) is isomorphic to the image of (f i ). Thus the sum ΣX () k i=1 Q,i(X) is a homology isomorphism ad therefore a homotopy equivalece. We ow wish to reproduce such wedge decompositios i the case of M (Y ), where Y is a homotopy coassociative co-h space. Let σ Σ. As i Sectio 3, let M (σ) be the composite M (σ) : M (Y ) s ΣX () Σσ ΣX () r M (Y ). Suppose f 1,, f k Z (p) [Σ ] is a orthogoal decompositio of the idetity. Suppose f i = Σ! j=1 a j σ j, where the σ j s are distict elemets of Σ ad each a j Z (p). Let M (f i ) : M (Y ) M (Y ) be the sum M (f i ) = Σ! j=1 a j M (σ j ). Sice the iclusio M (Y ) s ΣX () is a co-h map, Lemma 3.7 (a) implies: Lemma 4.2. There is a homotopy commutative diagram M (Y ) M(f i) M (Y ) s ΣX () f i s ΣX (). Let N i (Y ) = hocolim M(f i ) M (Y ). By Lemma 4.2 there is a homotopy commutative diagram of telescopes M (Y ) N,i (Y ) s ΣX () Q,i (Y ). 11

12 As M (Y ) s ΣX () is a co-h map, we ca add over i to get the homotopy commutativity of the diagram i the followig propositio. Propositio 4.3. There is a homotopy commutative diagram of equivaleces M (Y ) k i=1 N,i(Y ) s ΣX () k i=1 Q,i(X). Proof. It remais to show that the map e : M (Y ) k i=1 N,i(Y ) alog the top row is a homotopy equivalece. By defiitio, M (f i ) = Σ! j=1 a j M (σ j ). By Lemma 3.7 (d), (M (σ)) acts o H (M (Y )) = Σ(Σ 1 H (Y )) by permutig tesor factors. That is, (M (σ)) = Σσ. Thus (M (f i )) = Σf i. The telescope N,i (Y ) of the map M (f i ) therefore has its homology isomorphic to Im (Σf i ). Addig over i, we see that e has image isomorphic to that of Im ( Σ! j=1(σf i ) ). But f 1,..., f k is a orthogoal decompositio of the idetity so the latter image is isomorphic to that of the idetity map. Thus e is a isomorphism ad hece e is a homotopy equivalece. Next cosider the aturality of the homotopy decompositio M (Y ) k i=1 N,i(Y ). As i Sectio 3, let f : Y Z be a co-h map betwee homotopy coassociative co- H spaces. Let X = ΩY, X = ΩZ, g = Ωf, ad recall the defiitio of the map M (Y ) M(f) M (Z) (preceedig Lemma 3.4). Lemma 4.4. There is a homotopy commutative diagram M (Y ) M (f i ) M (Y ) M (Z) M (f) M (f i ) M (f) M (Z). Proof. To keep track of the space ivolved, deote the telescope maps M (Y ) ΣX () s r M (Y ) by s Y ad r Y respectively. Recall that r Y s Y is homotopic to the idetity map o M (Y ). Deote the map M (Y ) M(σ) M (Y ) by M Y (σ). Also write M Y (f i ) for M (Y ) M(f i) M (Y ). 12

13 Cosider the sequece: M Z (f i ) M (f) = ( Σ! j=1 a j M Z (σ j ) ) M (f) (1) Σ! j=1 ( aj M Z (σ j ) M (f) ) (2) Σ! j=1 ( aj M (f) M Y (σ j ) ) (3) ( Σ! j=1 M (f) a j M Y (σ j ) ) (4) M (f) ( ) Σ! j=1 a j M Y (σ j (5) = M (f) M Y (f i ). (6) The defiitio of M Z (f i ) gives the equality i lie (1). By Lemma 3.5, M (f) is a co-h map so it will distribute o the right, givig the homotopy i lie (2). The homotopy i lie (3) follows from Lemma 3.8. The commutatio of a j ad M (f) i lie (4) follows from the fact that M (f) is co-h. Ay map distributes o the left whe a sum is take via a co-h structure, givig lie (5). Fially, the equality i lie (6) comes from the defiitio of M Y (f i ). homotopies proves the lemma. This sequece of equalities ad Takig horizotal telescopes i Lemma 4.4 gives a homotopy commutative diagram M (Y ) N,i (Y ) M (Z) M (f) N,i (Z). Addig over i gives: Propositio 4.5. There is a homotopy commutative diagram of equivaleces M (Y ) k i=1 N,i(Y ) M (f) M N (Z) N,i (Z). The followig Theorem summarizes Propositios 4.3 ad

14 Theorem 4.6. Let Σ k i=1f i = 1 be a orthogoal decompositio of the idetity i Z (p) [Σ ]. To this there correspods a homotopy equivalece M (Y ) k N,i (Y ) which is atural for co-h maps Y Z betwee coassociative co-h spaces. i=1 Next, we examie the telescope maps N,i (Y ) Q,i (X) i Propositio 4.3 ad show they have atural left homotopy iverses. First, takig homotopy iverses i Propositio 4.3 ad restrictig to the i th wedge summad gives: Corollary 4.7. For each 1 i k, there is a homotopy commutative diagram N,i (Y ) k i=1 N,i M (Y ) Q,i (X) k i=1 Q,i(X) s ΣX (). where the left square is the iclusio of the i th summad ito the wedge. Corollary 4.8. For each 1 i k, the telescope map N,i (Y ) Q,i (X) has a left homotopy iverse. Proof. Sice M (Y ) s ΣX () ad N,i (Y ) M (Y ) each have left homotopy iverses, usig Corollary 4.7 we obtai a homotopy commutative diagram N,i (Y ) Q,i (X) M (Y ) s ΣX () r M (Y ) N,i (Y ) i which the upper directio is homotopic to the idetity map o N,i (Y ). Defie ψ,i : Q,i (X) N,i (Y ) by the composite ψ,i : Q,i (X) k Q,i (X) i=1 ΣX () r M (Y ) N,i (Y ). Each of the four maps i this composite is atural. The wedge iclusio ad the homotopy equivalece are atural with respect to maps X X. By Lemma

15 ad Propositio 4.5 respectively, the maps r ad M (Y ) N,i (Y ) are atural for co-h maps Y Z betwee coassociative co-h spaces. Thus: Lemma 4.9. The telescope map N,i (Y ) Q,i (X) has a left homotopy iverse Q,i (X) ψ,i N,i (Y ) which is atural with respect to co-h maps Y Z betwee homotopy coassociative co-h spaces. That is, there is a homotopy commutative diagram Q,i (X) ψ,i N,i (Y ) Q,i (X) ψ,i N,i (Z). 5. The costructio of A mi for coassociative co-h spaces Recall from Sectio 2 the idempotets λ max, δ max Z (p) [Σ ]. These give orthogoal decompositios of the idetity, 1 = λ max Recall that (X) = hocolim λ max (X), B max B max + (1 λ max ) ad 1 = δ max + (1 δ max ). ΣX (), (X) = hocolim 1 λ max ΣX (), ad (X) were defied similarly with respect to δ max. Let Y be a homotopy coassociative co-h space. As i Sectio 4, the elemet λ max a map M (λ max Let Z (p) [Σ ] correspods to ) : M (Y ) M (Y ), ad similarly for 1 λ max, δ max, ad 1 δ max M (Y ) = hocolim M(λ max ) M (Y ) M (Y ) = hocolim M(1 λ max ) M (Y ) MB max (Y ) = hocolim M(δ max ) M (Y ) MB max (Y ) = hocolim M(1 δ max ) M (Y ). Suppose Y Z is a co-h map betwee homotopy coassociative co-h spaces. As before, let X = ΩY ad X = ΩZ. By Theorem 4.6 ad Lemma 4.9 we have:. Lemma 5.1. The followig hold: (a) There is a homotopy equivalece M (Y ) M (Y ) M (Y ) which is atural for co-h maps Y Z betwee coassociative co-h spaces. 15

16 (b) There is a homotopy commutative diagram of equivaleces M (Y ) M (Y ) M (Y ) (X) (X) s ΣX (). (c) The map M (Y ) (X) i part (b) has a left homotopy iverse which is atural for co-h maps Y Z betwee homotopy coassociative co-h spaces. That is, there is a homotopy commutative diagram M (Y ) M (Z) (X) (X). The same statemets hold whe is replaced by B max. Now cosider what happes i homology. Recall from Sectio 2 that H ( (X)) = ( H (X)), where ( H (X)) is the image of the idempotet λ max : H (X) H (X). By Propositio 3.7 (d), the iclusio M (Y ) s ΣX () becomes a -fold tesor iclusio H (M (Y )) = Σ(Σ 1 H (Y )) (s ) H (ΣX () ) = Σ H (X). I geeral, if W V are vector spaces the the idempotet λ max (V ) : V V restricts to the correspodig idempotet o W, that is, λ max (V ) W = λ max (W ). So the image (V ) of λ max (V ) restricts to the image (W ) of λ max (W ). A similar argumet holds for the idempotet δ max to the case W = Σ 1 H (Y ) ad V = H (X) we have: (V ) ad its image B max (V ). Applied Lemma 5.2. There are isomorphisms H (M (Y )) = (Σ 1 H (Y )) H (MB max (Y )) = B max (Σ 1 H (Y )) 16

17 We ow begi the costructio of A mi (Y ) for ΩY. This is doe with the help of the kow costructio of A mi (X) for ΩΣX i the case whe X = ΩY. The two are liked by the co-h structure map Y Sectio 2 there is a homotopy fibratio A mi (X) s ΣΩY = ΣX i Lemma 3.1 (b). Recall from (X) π X ΣX, where (X) = =2 Qmax (X), ad π X is the wedge sum of the composites π,x : (X) ΣX () w ΣX. Similarly, let ad defie M (Y ) = =2 M (Y ) Mπ Y : M (Y ) Y by addig each of the composites Mπ,Y A mi (Y ) by the homotopy fibratio : M (Y ) M (Y ) w Y. Defie A mi (Y ) M (Y ) Mπ Y Y. Now we put together the two fibratios defiig A mi (Y ) ad A mi (X). Cosider the diagram M (Y ) M (Y ) w Y (X) s ΣX () w ΣX. The left square homotopy commutes by restrictig the diagram i Lemma 5.1 (b) to the left wedge summad. The right square homotopy commutes by Lemma 3.9. Addig over the gives a homotopy commutative diagram s M (Y ) Mπ Y Y (X) 17 π X s ΣX.

18 From this we obtai a homotopy fibratio diagram ΩM (Y ) ΩMπY ΩY A mi (Y ) M (Y ) Mπ Y Y Ω (X) Ωs Ωπ X ΩΣX A mi (X) (X) π X s ΣX. Note that alog the bottom row we have ΩΣX A mi (X) Ω (X). By Theorem 2.2, Ωπ X has a left homotopy iverse θ X : ΩΣX Ω (X) which is atural for maps X X. By Lemma 5.1 (c) the map M (Y ) (X) has a left homotopy iverse ψ,y : (X) M (Y ) which is atural for co-h maps Y Z betwee coassociative co-h spaces. Addig over we obtai a left homotopy iverse ψ Y : (X) M (Y ) of M (Y ) (X). Let Mθ Y be the composite Mθ Y : ΩY Ωs ΩΣX θ X Ω (X) Ωψ Y ΩM (Y ). Propositio 5.3. Mθ Y is a left homotopy iverse of ΩMπ Y. Thus ΩY A mi (Y ) ΩM (Y ). Proof. We have a homotopy commutative diagram ΩM (Y ) Ω (X) ΩY ΩMπ Y Ωs ΩΣX Ωπ X θ X Ω (X) ΩψY ΩM (Y ) i which the lower row is the defiitio of Mθ Y ad the upper directio is the idetity map o ΩM (Y ). Remark 5.4. The left homotopy iverse for Ωπ X is costructed explicitly i [SW1, SW2] by usig combiatorial James-Hopf ivariats. It may be possible to reproduce combiatorial James-Hopf maps for ΩY usig the methods i Sectio 3, but it is certaily more efficiet to proceed as above by the timely use of existig retractios. 18

19 The ext propositio shows that A mi (Y ) is the smallest atural coalgebra retract of ΩY. Propositio 5.5. H (A mi (Y )) = A mi (Σ 1 H (Y )). Proof. By Theorem 2.1 there is a coalgebra decompositio H (ΩY ) = T (Σ 1 H (Y )) = A mi (Σ 1 H (Y )) B max (Σ 1 H (Y )). The homotopy decompositio i Propositio 5.3 implies the statemet we are tryig to prove is equivalet to showig that H (ΩM (Y )) = B max (Σ 1 H (Y )). But by defiitio, M (Y ) = =2 MQmax H (ΩM (Y )) = H (Ω( =2 (Y ) so by Lemma 5.2 we have M (Y ))) = T ( =2 (Σ 1 H (Y )) = B max (Σ 1 H (Y )). It remais to show that the decompositio i Propositio 5.3 is atural. Suppose Y Z is a co-h map betwee coassociative co-h spaces. The aturality i Lemmas 5.1 (a) ad 3.9 give, for each 2, a homotopy commutative diagram M (Y ) M (Y ) w Y M (Z) M (Z) w Z. Addig over gives a homotopy commutative diagram M (Y ) MπY Y M (Z) MπZ Z. From this we obtai a homotopy fibratio diagram ΩM (Y ) ΩMπY ΩY A mi (Y ) M (Y ) MπY Y ΩM (Z) ΩMπY ΩZ A mi (Z) M (Z) MπZ Z. 19

20 First observe: Lemma 5.6. The left homotopy iverse Mθ Y of ΩMπ Y i Propositio 5.3 is atural with respect to co-h maps Y Z betwee homotopy coassociative co-h spaces. That is, there is a homotopy commutative diagram ΩY MθY ΩM (Y ) ΩZ MθZ ΩM (Z). Proof. As usual, we have X = ΩY, X = ΩZ, ad g : X X is Ωf. Lemma 3.1 (c) there is a homotopy commutative diagram By Y ΣX Z f ΣX Σg i which all maps are co-h maps. Cosider the diagram ΩY ΩΣX θx Ω (X) ΩψY ΩM (Y ) ΩZ Ωf ΩΣg ΩΣX θz Ω (X) ΩψZ ΩM (Z). The left square homotopy commutes by the previous diagram. The middle square homotopy commutes by the aturality of Theorem 2.2. defiitio (preceedig Propositio 5.3), ψ Y For the right square, by = =1 ψ,y, where each ψ,y is atural for co-h maps Y Z betwee coassociative co-h spaces. Thus the right square homotopy commutes as well ad hece the etire rectagle homotopy commutes. But the top row of the diagram is the defiitio of Mθ Y defiitio of Mθ Z. while the bottom row is the Propositio 5.7. The homotopy decompositio ΩY A mi (Y ) ΩM (Y ) of Propositio 5.3 is atural with respect to co-h maps Y Z betwee homotopy coassociative co-h spaces. 20

21 Proof. The homotopy fibratio diagram preceedig Lemma 5.6 gives a homotopy commutative square ΩY ΩZ A mi (Y ) A mi (Z). Combiig this with the diagram i Lemma 5.6 gives a homotopy commutative diagram of equivaleces ΩY A mi (Y ) ΩM (Y ) ΩZ A mi (Z) ΩM (Z), which proves the Propositio. Fially, observe that Theorem 1.1 is the combiatio of Propositios 5.3, 5.5, ad 5.7. Refereces [G] T. Gaea, Cogroups ad suspesios, Ivet. Math. 9 (1970), [SW1] P. Selick ad J. Wu, O atural coalgebra decompositios of tesor algebras ad loop suspesios, Mem. Amer. Math. Soc. 148, No. 701 (2000). [SW2] P. Selick ad J. Wu, The fuctor A mi o p-local spaces, preprit. [SW3] P. Selick ad J. Wu, O fuctorial decompositios of self-smash products, preprit. [T] S.D. Theriault, Homotopy decompositios ivolvig the loops of coassociative co-h spaces, to appear i Caad. J. Math. Deparmet of Mathematics, Uiversity of Toroto, Toroto, ON, M5S 3G3, Caada address: selick@math.toroto.edu Departmet of Mathematical Scieces, Uiversity of Aberdee, Aberdee AB24 3UE, Uited Kigdom address: s.theriault@maths.abd.ac.uk Departmet of Mathematics, Natioal Uiversity of Sigapore, Sigapore , Republic of Sigapore, address: matwuj@us.edu.sg 21