J. Klein, R. Schwanzl, R. M. Vogt. It has been known for a long time that an (n? 1)-connected CW -complex

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1 Co-Mltiplication and Sspension J. Klein, R. Schanzl, R. M. Vogt 1 Introdction It has been knon for a long time that an (n? 1)-connected CW -complex of dimension 2n? 1 has the homotopy type of a sspension. In 1963 Bernstein and Hilton proved that an (n? 1)-connected based CW -complex X of dimension 3n? 3 has the homotopy type of a sspension provided X admits a co-mltiplication X! X _X ith homotopy co-nit [1] (Bernstein and Hilton made the additional technical reqirement that the homology grops of X are nitely generated). In 1970 Ganea extended this reslt: An (n? 1)-connected co-h-space X of dimension n? 5, n 2, has the homotopy type of a sspension Y if and only if it is homotopy co-associative. Moreover, the homotopy eqivalence Y! X is a homomorphism p to homotopy [5]. Bernstein-Hilton and Ganea conjectred that a sitable theory of co-a n -spaces, dal to Stashe's theory of A n -H-spaces [12], old lead to extensions of their reslts. Indeed, in the A n -space sitation the folloing reslt holds [13], [16]. 1.1 Proposition: Let X be an (n? 1)-connected ell-pointed A k?1 -space, k 3, admitting a homotopy inverse, sch that i X = 0 for i > (k +1)n?. Then there is an A k?1 -homomorphisms f : X! M Y hich is a eak eqivalence. Here M Y is the Moore loop space on Y ith its natral monoid strctre. Recall that an A k -homomorphism is a homomorphism p to homotopy satisfying coherence conditions p to (k? 1)-dimensional homotopies. The prpose of this paper is to prove the folloing dal reslt. 1

2 1.2 Theorem: Let X be an (n? 1)-connected based CW -complex ith dim X k(n? 2) + 3, n 2. Then X is of the homotopy type of a sspension if and only if X is a co-a k?1 -space. Moreover, the homotopy eqivalence f : Y! X is a co-a k?1 -homomorphism. 1.3 Corollary: A 2-connected space X is of the eak homotopy type of a sspension if and only if it is a co-a 1 -space. We then se or approach to prove a dal version of Segal's delooping reslt [11] 1. Theorem: Let X :! T op be a co-simplicial space sch that X 0 is contractible, X 1 is 2-connected and ( 1 ; : : : ; n ) : _ n k=1 X 1! X n is a homotopy eqivalence here k : [1]! [n] maps 0 to i? 1 and 1 to i. Then there is a eak homotopy eqivalence RX! X 1, here RX is a fnctorial, thickened version of the topological realization of X. A reslt of this type has been formlated by Hopkins [7] ith the eaker condition that X 1 is only 1-connected, bt remarks by Boseld [3, (.9)] indicate that additional assmptions are needed. Instead of the maps k Hopkins ses the eqivalence ( 1 ; : : : ; n ) : _ n X ' k=1 1?! X n here k : [1]! [n] maps 0 to 0 and 1 to k. A cosimplicial space in his sense indces one in or sense by a base change. The approach to despension hich e develop here is motivated by potential applications. A brant variant of the main reslt of this paper (see [8], [9]) ill be sed by the rst-named athor in a sbseqent paper to prove a Haeiger-type embedding theorem in the category of Poincare dality spaces. Sch embedding and desspension reslts also play a fndamental role in the original Poincare srgery program as formlated by Broder et. al. It is hoped that the reslts of this paper can be adapted so as to complete the Broder program. Or proofs ill make extensive se of homotopy limits and co-limits of cbe diagrams and their analysis by Goodillie [6]. They are in spirit qite dierent from the proofs of Bernstein-Hilton and Ganea. To stay aay from point 2

3 set topological diclties e ork in the category T op of compactly generated spaces in the sense of [1]. Based spaces are assmed to be ell-pointed nless stated otherise. 2 A k -spaces and co-a k -spaces We recall the denition of an A k -space sing the terminology of [2]. Let G be the category ith ob G = = f0; 1; : : :g, exactly one morphism n : n! 1 for n 1, and G(n; k) consisting of all formal expressions i1 : : : ik ; i 1 + : : : + i k = n G(0; 0) = fid 0 g and G(0; n) = ; = G(n; 0) for n > 0. Composition is dened by the reqirement that is a bifnctor. The category G describes semigrops: A semigrop G determines a fnctor G! T op, n 7! G n, transforming into. Conversely, if G : G! T op is a fnctor sch that G(n) = G(1) n and G(f g) = G(f) G(g) then G(1) is a semigrop ith mltiplication G( 2 ). We symbolize n ; n 2, by a box ith n inpts x 1 ; : : : ; x n and one otpt x 1 : : : x n. Composite operations are obtained by iring boxes together to a tree shaped circit, e.g. 2.1 x y y z z x y z x (1) (2) (3) (xy)z xyz x(yz) represent all possible non-trivial composite operations 3! 1. In G they coincide bt for homotopy associative mltiplications they only coincide 3

4 p to homotopy. To accont for this e give each connection (beteen boxes) a length t 2 I = [0; 1]. Given a tree T the varios lengths of its connections make p the points of a cbe C(T ) hose dimension is the nmber of connections of T. We ths obtain a ne category T G ith ob T G = ob G and T G(n; 1) = ` C(T ), T rnning throgh all trees ith n inpts, T G(n; k) = ` T G(i 1 ; 1): : :T G(i k ; 1) ith i 1 +: : :+i k = n. Let T be a tree ith lengths in T G(k; 1) and let (T 1 ; : : : ; T k ) 2 T G(n; k) be a k-tple of trees then T (T 1 ; : : : ; T k ) is the tree obtained from T by sticking T i onto its i-th inpt and giving the nely created connections the lengths 1. The trivial tree serves as id 1. We have a continos pairing : T G T G! T G dened by (T 1 ; : : : ; T k )(T k+1 ; : : : ; T k+l ) = (T 1 ; : : : ; T k+l ). Composition then is determined by the reqirement, that is a bifnctor. It is continos. Let WG be the qotient category of T G obtained by imposing the relation that a connection of length 0 may be shrnk by amalgamating the to boxes at its ends to a single one. Observe that the trees (2.1.1) and (2.1.3) represent nit intervals in WG(3; 1) hich are identied at 0 ith the tree (2.1.2). Hence the trees of (2.1) codify an associating homotopy. 2.2 Denition: Let W k G be the fll sbcategory of WG of objects 0; 1; : : : ; k. An A k -space is a continos fnctor X : W k G! T op mapping to (in particlar X(n) = X(1) n ) sch that the mltiplication X( 2 ) : X(1) 2! X(1) admits a homotopy nit. (ote that e do not reqire any coherence for the homotopy nit.) In abse of notation e often denote X(1) by X. 2.3 Remark: The map " : WG! G hich maps a tree ith n inpts to n is a continos fnctor admitting a non-fnctorial section : G! WG mapping n to the tree ith a single vertex and n inpts if n > 1 and to the trivial tree for n = 1. As maps of morphism spaces " ' id (the homotopy shrinks the lengths of connections by the factor t at time t). It is easy to sho that WG(n; 1) is a sbdivision of the Stashe cell K n of [12] into cbes. So or denition coincides ith the one of Stashe ith the exception that he makes the stronger reqirement that X( 2 ) admits a strict nit e. Bt if feg X is a closed co-bration both denition are eqivalent [2, Chapt. I]. co-a k -spaces are dened dally:

5 2. Denition: A co-a k -space is a continos fnctor X : (W k G) op! T op into the category of ell-based topological spaces mapping to the edge sm _ sch that the comltiplication X( 2 ) : X(1)! X(1) _ X(1) admits a homotopy co-nit (i.e. p i X( 2 ) ' id, here p i : X(1) _ X(1)! X(1) is the i-th projection, i = 1; 2). Again e often rite X for X(1). 2.5 Examples: (1) Let C be an operad ithot permtations in the sense of [10]. We associate a category C ith ob C =, C(n; k) = ` C(i 1 ): : :C(i k ) ith i 1 +: : :+i k = n and all i j > 0. C(0; 0) = fid 0 g and C(0; n) = C(n; 0) = ; if n > 0. Taking prodcts denes a pairing : C C! C. Composition C(k; 1) C(n; k)?! C(n; 1) is dened by the strctre map of the operad. General composition then is determined by the reqirement that is a bifnctor. Call C an A k -operad if C(j) ' for j k. If C is an A k -operad, there is a continos -preserving fnctor F C : W k G! C by [2, 3.17]. If F 0 C is another sch fnctor then there is a homotopy throgh continos - preserving fnctors from F C to F 0 C. A co-c-space is a collection of maps n : X ^ C(n) +?! X _ : : : _ X = X _n (n-fold edge) here C(n) + = C(n) t f+g, sch that 1 (x; 1) = x and X ^ C(k) + ^ C(i 1 ) + ^ : : : ^ C(i k ) + = X ^ (C(k) C(i 1 ) : : : C(i k )) + id ^ X ^ C(n) + k ^ id X _k ^ C(i 1 ) + ^ : : : ^ C(i k ) + n proj (X ^ C(i 1 ) + ) _ : : : _ (X ^ C(i k ) + ) X _n i1 _ : : : _ ik commtes. It determines a co-c-space, i.e. a continos fnctor C op! T op mapping to _. 5

6 If C is an A k -operad, k 2, then any co-c-space has a homotopy co-nit. Hence it is a co-a k -space. (2) Let Q be the \little 1-cbes" category [2, 2.53] describing loop space strctres. A point in Q(n; 1) is a tple (x 1 ; y 1 ; : : : ; x n ; y n ) of points in I sch that 0 x 1 < y 1 x 2 < y 2 : : : x n < y n 1, and shold be considered as an embedding of n intervals [x i ; y i ] into I sch that the images have disjoint interiors. Let Z be a based space. Then Z is a co-q-space and hence a co-a 1 -space ith n : Z ^ Q(n; 1) +?! Z _ : : : _ Z 8 < z; t? x k 2 k-the smmand if t 2 [x n ((z; t); (x 1 ; y 1 ; : : : ; x n ; y n ) = k ; y k ] y : k? x k otherise 3 Cbical diagrams of co-a k -spaces Let D be a small category. Homotopy coherent D-diagrams are codied by the topological category WD [2, Chapt. VII] ith ob WD = ob D and WD(A; B) = a n D n+1 (A; B) I n! = here D n+1 (A; B) is the set of (n + 1)-tples (f n ; : : : ; f 0 ) of morphisms in D sch that f n : : : f 0 : A! B is dened, ith the relations (f n ; t n ; : : : ; f 1 ; t 1 ; f 0 ) = 8 >< >: Composition is dened by (f n?1 ; t n?1 ; : : : ; f 0 ) if f n = id (f n ; : : : ; f i+1 ; max(t i+1 ; t i ); f i?1 : : : ; f 0 ) if f i = id (f n ; t n ; : : : ; t 2 ; f 1 ) if f 0 = id (f n ; : : : ; f i+1 f i ; t i : : : ; f 0 ) if t i+1 = 0: (f n ; t n ; : : : ; f 0 ) (g p ; p ; : : : ; g 0 ) = (f n ; t n ; : : : ; f 0 ; 1; g p ; p : : : ; g 0 ) WD can be vieed as obtained from D by taking the free category over the diagram D and ptting back the relations p to coherent homotopies. 6

7 3.1 Lemma: The agmentation " : WD! D; (f n ; t n ; : : : ; f 0 ) 7! f n : : :f 0 is a homotopy eqivalence of morphism spaces ith non-fnctorial homotopy inverse : D! WD; f 7! (f). [2, Prop. 3.15] 3.2 Denition: [2] A homotopy coherent D-diagram is a continos fnctor D : WD! T op. 3.3 otation: Q n denotes the poset of all sbsets of n = f1; : : : ; ng, J n denotes the poset of all sbsets of n dierent from n T n denotes the poset of all non-empty sbsets of n We are going to associate to each co-a k?1 -space a homotopy coherent T k diagram k 2. Let Inj(n; k) denote the set of ordered injections n! k. We also allo 0 = ;. We enlarge WG by set operations coming from injections (compare [2, p. 57 ]) to a category W G as follos: ob W G = ob WG and W G(n; k) = a n WG(l; k) Inj(l; n) In particlar, W G(n; 0) consists of a single element (id 0 ; ;! n). Composition of (f; ) 2 WG(l; k) Inj(l; n) ith (g 1 : : : g n ; ) 2 WG(m; n) Inj(m; p) is dened by (f; ) (g 1 : : : g n ; ) = (f (g (1) : : : g (l) ); (m 1 ; : : : ; m n )) Here g i 2 WG(m i ; 1); m = m 1 + : : : + m n, and (m 1 ; : : : ; m n ) : m (1) + : : : + m (l)! m is the injection sending m (1) +: : :+m (i?1) +j to m 1 +m 2 +: : :+m (i)?1 +j for 1 j m (i). The bifnctor extends to W G by (f; ) (g; ) = (f g; t ) here t is the ordered disjoint nion. 7

8 Let W k G again denote the fll sbcategory of W G consisting of all objects j k. Given a co-a k -space X : (W k G) op! T op it extends to a continos fnctor X : (W kg) op?! T op mapping to _ by X : (f; ) : X _k f?! X _l?! X _n for (f; ) 2 WG(l; k)inj(l; n), here maps the i-th smmand identically onto the (i)-th one (the checking of fnctoriality is left to the reader). In an analogos ay e can adjoin the injections to G to obtain a category G. We have the folloing fact. 3. The agmentation fnctor " : WG! G and its section : G! WG extend to a fnctor " : W G! G and a non-fnctorial map : G! W G. Again " is a homotopy eqivalence on morphism spaces ith homotopy inverse. 3.5 Let inj denote the category of ordered sets [n] = f0; 1; : : : ; ng and order preserving injections and inj k the fll sbcategory of objects [j]; j k. Dene a fnctor # k : T k?! inj k?1 by # k (fi 1 ; : : : ; i p g) = [p? 1]. Identify i 1 < : : : < i p in order ith 0 < 1 < : : : < p? 1. Then an inclsion : fi 1 ; : : : ; i p g fj i ; : : : ; j q g denes a niqe order preserving map # k () : [p? 1]! [q? 1]. (We alays list the elements of a sbset of n in increasing order.) 3.6 ' k : inj k?! (G k )op is dened by sending [j] to j and : [p]! [q] to pl i=1 (i)?(i?1) ;, here : (p)? (0)! q is the ordered injection missing the elements f1; : : : ; (0); (p) + 1; : : : ; qg 3.7 Let X : (W k?1 G) op! T op be a co-a k?1 -space. To simplify or ftre argment e explicitly constrct the 2-faces of a homotopy T k -diagram k : WT k k?! (W k?1 G)op X?! T op 8

9 before e apply the niversal property of WT k. Since k factors throgh X it is fnctorial in X. On objects k is the composite ' k?1 # k, and on morphisms (f) it is given by ' k?1 # k (f), here f 2 T k and : T k! WT k, : (G k?1 )op! (W k?1 G ) op are the sections. Finally e specify the \homotopies" of 2-dimensional faces dened by fi 1 ; : : : ; i p g fj 1 ; : : : ; j p?2 g in T k. Let r : 1! 2 denote the injection missing the r-th element. If j 1 < j 2 < i 1 the face is mapped to the commtative sqare (3.7.1) id p If j 1 < i 1 < i r < j q < i r+1 the face is mapped to the commtative sqare (3.7.2) r 1 id r?1 r + 1 id r?1 2 id r 2 id p?r?1 r id r r + 2 If i r < j q < i r+1 < i s < j t < i s+1 the face is mapped to the commtative sqare (3.7.3) s? r id s?r s? r + 2 id r?1 id s?r 2 id s?r+1 2 id p?s?1 s? r + 2 s? r id s?r+1 If i r < j q < j q+1 < i r+1 the face is mapped to the diagram 9

10 (3.7.) id r? id id p?r?1 2 3 id 2 and the associating homotopies in Wk?1G. The faces ith i r < i p < j p+1 and i p < j p+1 < j p+2 are mapped analogosly to (3.7.2) and (3.7.1) ith 1 replaced by 2 and the orders ith respect to reserved. Having specied k on the 2-dimensional faces e extend it over all of WT k by applying the relative lifting theorem [2, 3.17]. 3.8 Lemma: Let X be a 1-connected co-a k?1 -space, k 3. Then for each (k? 1)-dimensional face S of T k the homotopy coherent S-diagram kjws : WS?! T op is strongly (homotopy) co-cartesian in the sense of [6, 2.1]. Proof: We have to sho that the specied faces of the fnctor k are homotopy pshots after composition ith X. This is trivially tre for (3.7.2) and (3.7.3): Given maps f : A! B and g : C! D in T op, the sqare A _ C f _ id B _ C id _ g id _ g f _ id A _ D B _ D is alays a homotopy pshot. For (3.7.1) and the corresponding one ith 1 replaced 2 the reslt holds by [18, Lemma 3.3]. For (3.7.) e se the fact that a 1-connected homotopy associative co-h-space hich is a CW -complex has a 2-sided inverse [5, Prop.3.6]. 2 10

11 Proof of Theorem Lemma: The reslts of Ellis-Steiner [] and Goodillie listed in [6, Thm. 2.3,..., Thm. 2.6], hich are formlated for commtative cbe diagrams, also apply to coherently homotopy commtative cbe diagrams. (A stronger version of [6, Thm. 2.] for homotopy commtative cbe diagrams can be fond in [17, Prop. 5.5]). Proof: By [15, Prop. 5.] each coherently homotopy commtative cbe is eqivalent to a strictly commtative one, and the eqivalence indces an eqivalence of their homotopy limits and colimits. 2.2 Constrction: Let l : WT l! T op, 2 l k be the cbe diagrams determined by the co-a k?1 -space X. Let S k?1 T k be the face dened by all sbsets of k containing 1, and T S k?1 obtained from it by deleting the initial vertex. Let P l denote the homotopy limit (in the sense of [15]) of the coherently homotopy commtative T l -diagram l and R k?1 the one of the sbdiagram WT S k?1! T op. Observe that P l and R k?1 are ell-pointed by [15, Prop. 6.9]. Then P k is the homotopy limit of the indced diagram = k f1g R k?1 P k?1 (compare [17, Prop. 5.]). In particlar, e have a seqence of brations P k p?! k p k?1 p Pk?1?!?! 3 P2 = X ith ber(p k ) = R k?1. Since k jws k?1 is strongly co-cartesian and each map k (knfig! k) : X _(k?2)! X _(k?1) is (n? 1)-connected the indced map! R k?1 is (1?(k?1)+(k?1)(n?1)) = (1+(k?1)(n?2))-connected. Since n 2, R k?1 is at least 1-connected and hence each P k connected. P k together ith the niversal map P k! k extends k to a WQ k -diagram k : WQ k! T op, i.e. to a homotopy coherent k-dimensional cbe diagram. Let T ~ Q k be an l-dimensional terminal face, l < k. By (3.8) it is strongly co-cartesian, and hence by [] (1 + l(n? 2))-Cartesian. Since k is innitely Cartesian it is (k? k(n? 2))-co-Cartesian by [6, Thm. 2.6]. Recall that J l Q k is the sbcategory of all sbsets of l k, l k, except of l 11

12 itself. Let I l?1 J l denote the sbcategory of all sbsets containing 1 and J I l?1 the fll sbcategory of I l?1 containing all bt the terminal element. Let M l denote the homotopy colimit of k jwj l. By (3.8) the homotopy colimit of k jwj I l?1 is X _(l?1) and the indced map to the terminal vertex of I l?1 is a homotopy eqivalence. By [17, Prop. 5.] e have homotopy pshot diagrams.3 M l?1 r l?1 _(l?2) X q l X _(l?1) g l Ml for 3 l k, here r l : M l?! X _(l?1) is the indced map into the terminal vertex of k jwq l. Then, as noted above, r l g l is a homotopy eqivalence. Since k is (k(n? 2) + k + 1)-co-Cartesian, r k is (k + k(n? 2))-connected, hence g k is (k(n?2)+k)-connected. Since (.3) is a homotopy pshot, r k?1 is homology-(k(n? 2) + k)-connected. By donards indction e obtain r 2 : P k = M 2?! X is homology (k(n?2)+3)-connected. Since P k and X are both 1-connected, this implies that r 2 is (k(n? 2) + 3)-connected. The nal diagram.3 is obtained from the cbe P k 6 X ''' X ') X 6 X _ X By denition, q l factors throgh the standard pinch map and a edge so that e arrive at a homotopy commtative diagram 12

13 P k pinch P k _ P k f _ g r 2 X' '' ' '' ') g 3 r 3 M3 X _ X X _ X Since r 3 g 3 ' id and the pinch map and have co-nits e dedce f ' r 2 ' g. Hence r 2 is a homomorphism of co-h-spaces p to homotopy (to obtain higher coherence one has to inclde the higher dimensional cbes into the argment; e leave this to the reader). We smmarize. Proposition: Let X be an (n? 1)-connected co-a k?1 -space, n 2, k 3. Let P k be the homotopy limit of its associated coherently homotopy commtative T k -diagram. Then the indced map r : P k?! X is a (k (n? 2) + 3)-connected co-a k?1 -homomorphism. Proof of the Theorem 1.2: Let K be the CW -approximation of P k, and sppose that X is a CW -complex ith dim X k(n? 2) + 3. Since H k(n?2)+3 X is free abelian and r indces a map q : K! X hich is a homology isomorphism in dimensions less than k(n? 2) + 3 and a homology epimorphism in dimensions k(n? 2) + 3, there is a CW -complex Y having the same (k(n? 2) + 1)-skeleton as K by [1, Thm. 2.1] and a homology isomorphism Y! X. Since Y and X are 1-connected this is a homotopy eqivalence. 5 Proof of Theorem 1. Let X :! T op be a based cosimplicial space sch that X 0 is contractible, X 1 is 2-connected, and ( 1 ; : : : ; n ) : Dene : X nw k=1 X 1! X n is a homotopy eqivalence.?! d1 X 2?! X _ X 13

14 here the second map is a homotopy inverse of ( 1 ; 2 ) = (d 2 ; d 0 ). (In abse of notation e rite X for X 1 ). Then (X; ) is a co-h-space becase X d s 1 X (d 2 ; d 0 ) proj 1 X X _ X (d 2 ; d 0 ) h hh d 1 X 2 h hh proj 1 commtes p to homotopy since X 0 is contractible. In a similar ay one veries that (X; ) is homotopy associative. 5.1 Lemma: Each (k? 1)-dimensional face of X # k : T k! inj k?1! T op is strongly homotopy co-cartesian. Proof: The 2-dimensional face determined by : fi 1 ; : : : ; i p g fj 1 ; : : : ; j p+2 g is mapped to X p?1 d r?1 X p d q X p d r d q X p+1 if misses j q+1 < j r+1. For 0 < i < p + 1 consider X i?1 _ (X _ X) _ X p?i id id Q h hj X s 0 X i?1 _ X _ X p?i id _ d 1 _ id id _ (d 2 ; d 0 ) _ id Xi?1 _ X 2 _ X p?i ' X p d i ' X p+1 here the horizontal eqivalences are indces by the inclsions of [i? 1], the map i respectively ( i ; i+1 ) and the injections of [p? i] onto the terminal parts of [p] respectively [p + 1]. The sqare commtes hile the triangle 1

15 commtes p to homotopy. For d 0 and d p+1 the corresponding sqares are X _ X p incl: X incl: p X p _ X ( 1 ; d 0 ) ' ' (d p+1 ; p+1 ) X p+1 d 0 X d p+1 p X p+1 Hence, p to identities on edge smmands, e arrive at the same diagrams as in the proof of Theorem Proof of Corollary 1.3 and Theorem 1.: Let P k denote the homotopy limit of X # k. As in the proof of Theorem 1.2 e obtain a seqence of brations p?! p P?! 3 P3?! P2 ' X sch that p k : P k! P k?1 is k-connected becase X is 2-connected, and a (k + 3)-connected map r k : P k! X. Let P be the homotopy limit of the P k. Since lim 1 i (P k ) = 0 e have k (P ) = lim k (P i ) = k (P k ): In particlar, the natral projection q k : P! P k is k-connected. The maps r : P k! X are compatible ith the p k p to homotopy and indce a map P?! X: Since P! P k! X is (k + 1)-connected, this map is a eak eqivalence. References [1] I. Bernstein, P.J. Hilton, On sspension and comltiplications, Topology 2 (1963), [2] J.M. Boardman, R.M. Vogt, Homotopy invariant strctres on topological spaces, Springer Lectre otes in Math. 37 (1973). [3] A.K. Boseld, On the homology spectral seqence of a cosimplicial space, Amer. J. Math. 109 (1987),

16 [] G. Ellis, R. Steiner, Higher-dimensional crossed modles and the homotopy grops of (n + 1)-ads, J. Pre Appl. Algebra 6 (1987), [5] T. Ganea, Cogrops and sspensions, Inventiones math. 9 (1970), [6] T. Goodillie, Calcls II: Analytic fnctors, K-Theory 5 (1992), [7] M.J. Hopkins, Formlations of cocategory and the iterated sspension, Asterisqe (198), [8] J.R. Klein, Embedding theory in the Poincare category, in preparation. [9] J.R. Klein, R. Schanzl, R.M. Vogt, Parametrized sspension theory, in preparation. [10] J.P. May, The geometry of iterated loop spaces, Springer Lectre otes in Math. 271 (1972). [11] G. Segal, Categories and cohomology theories, Topology 13 (197), [12] J.D. Stashe, Homotopy associativity of H-spaces I, Trans. Amer. Math. Soc. 108 (1963), [13] J.D. Stashe, H-spaces from the homotopy point of vie, Springer Lectre otes in Math. 161 (1970). [1] R.M. Vogt, Convenient categories of topological spaces for homotopy theory, Arch. Math. 22 (1971), [15] R.M. Vogt, Homotopy limits and colimits, Math. Z. 13 (1973), [16] R.M. Vogt, A remark on A n -spaces and loop spaces, J. London Math. Soc. (2), 1 (1976), [17] R.M. Vogt, Commting homotopy limits, Math. Z. 153 (1977), [18] M. Walker, Homotopy pll-backs and applications to dality, Can. J. Math. 29 (1977),