On Maps from Loop Suspensions to Loop Spaces and the Shuffle Relations on the Cohen Groups. J. Wu

Transcription

1 On Maps from Loop Suspensions to Loop Spaces and the Shuffle Relations on the Cohen Groups J. Wu Author address: Department of Mathematics, National University of Singapore, Singapore , Republic of Singapore address: URL:

2 Contents Introduction 1 Chapter 1. Maps from Loop Suspensions to Loop Spaces The James Construction Bi- -groups Skeletons of Bi- -groups The Cohen Construction 30 Chapter 2. Shuffle Relations Functors to Coalgebras Geometric Realizations and the Shuffle Relations Proof of Theorem Proof of Theorem 4 55 Bibliography 61 v

3 Abstract The maps from loop suspensions to loop spaces are investigated using group representations in this article. The shuffle relations on the Cohen groups are given. By using these relations, a universal ring for functorial self maps of double loop spaces of double suspensions is given. Moreover the obstructions to the classical exponent problem in homotopy theory are displayed in the extension groups of the dual of the important symmetric group modules Lie(n), as well as in the top cohomology of the Artin braid groups with coefficients in the top homology of the Artin pure braid groups Mathematics Subject Classification. Primary 55P; Secondary 18G20, 20C05,55Q, 55U. Key words and phrases. loop spaces, James construction, Hopf invariants, bi- -groups, Cohen groups, functors to coalgebras, shuffle relations. Received by the editor January 25, Research is supported in part by the Academic Research Fund of the National University of Singapore. vi

4 Introduction 1 In [9, 10, 11], Cohen introduced a combinatorial group H with a representation to the group of the homotopy classes of functorial self maps of loop suspensions. This group is important for studying the classical exponent problem in homotopy theory, for instance the classical results in [2] can be obtained by some simple combinatorial computations in the group H. In [39, 40, 41, 45, 48], the group H has been successfully applied to solve some problems in the classical homotopy theory including the Cohen conjecture. As a combinatorial group, H has connections with homotopy string links studied by Milnor and Habegger-Lin in low dimensional topology [27, 28, 19, 23], as well as braid groups and simplicial groups [3, 13, 46]. The purpose of this article is to study the maps from loop suspensions to loop spaces using group representations. In particular, we obtain a generalization of the Cohen group H. Our generalization provides a way to construct various Cohen type groups related to maps from loop suspensions to loop spaces. By investigating the reduced diagonals, we obtain the shuffle relations on the Cohen groups. The quotient of the Cohen group H by the shuffle relations gives a universal ring R for functorial self maps of double loop spaces of double suspensions. The ring R is related to the extension groups of the important symmetric group modules Lie(n) by investigating functors to coalgebras. Moreover the obstructions to the exponent problem in homotopy theory are displayed in these extension groups. In addition to homotopy theory, the representation theory of the ring R is related to the functorial version of the Poincaré-Birkhoff-Witt Theorem, with connections to the modular representation theory of the symmetric groups. The remainder of this introduction describes our results in more details. Let X be a pointed space. Recall that the James construction J(X) is the free monoid generated by points in X modulo the single relation that the basepoint = 1, with the weak topology. The classical James theorem [21] states that J(X) is (weak) homotopy equivalent to ΩΣX if X is path-connected. The James filtration {J n (X)} is the word filtration, namely J n (X) is the quotient space of X n by the equivalence relation generated by (x 1,..., x i 1,, x i,..., x n 1 ) (x 1,..., x j 1,, x j,..., x n 1 ) for any 2 i, j n. Let X (n) denote the n-fold self smash product of X. Then J n (X)/J n 1 (X) is homeomorphic to X (n). Let q n : X n J n (X) be the quotient map. By the suspension splitting theorem [21], the inclusions J n 1 (X) J n (X) induce a tower of group epimorphisms [J(X), ΩY ] [J n (X), ΩY ] [X, ΩY ] 1 Received by the editor January 25, Research is supported in part by the Academic Research Fund of the National University of Singapore. 1

5 2 INTRODUCTION with the inverse limit [J(X), ΩY ] = lim n [J n (X), ΩY ] and group monomorphism q n : [J n (X), ΩY ] [X n, ΩY ] for each n 1. Our study on the group of homotopy classes [J(X), ΩY ] is given by introducing operations on the sequence of groups {[X n+1, ΩY ]} n 0 described as follows. Note that the group [J(X), ΩY ] = [ΩΣX, ΩY ] if X is a path-connected CW -complex. The coordinate inclusions and projections d i : X n+1 X n (x 0, x 1,..., x n ) (x 0,..., x i 1, x i+1,..., x n ) d i : X n X n+1 (x 0, x 1,..., x n 1 ) (x 0,..., x i 1,, x i,..., x n 1 ) induce functions (group homomorphisms if Z is a homotopy associative H-space with inverse) d i = d i : [X n+1, Z] [X n, Z] and d i = d i : [X n, Z] [X n+1, Z] for any pointed space Z with the following identities: d j d i = d i d j+1 for j i, d j d i = d i+1 d j for j i, d i 1 d j for j < i, d j d i = id for j = i, d i d j 1 for j > i, where d 0 x = for x [X, Z]. The sequence of sets K(X, Z) = {[X n+1, Z]} n 0 with the operations d i and d i has the property that K(X, Z) is a -set under the faces given by d i and a co- -set under the cofaces given by d i with the third identity mixing faces and cofaces together. The third identity is different from simplicial identities. Motivated from the above three identities, we introduce the concept of bi- -set (bi- -group), namely, a sequence of sets (groups) with faces (face homomorphisms) and cofaces (coface homomorphisms) satisfying the above identities. The face operations are important for understanding [J(X), Z] by the following lemma. On the other hand, coface operations are used for constructing Hopf invariants. Let S = {S n } n 0 be a -set, the Cohen set (Cohen group if S is a -group) H n S is defined to be the equalizer of the faces, namely H n S = {x S n d 0 x = d 1 x = = d n x}. The total Cohen set (total Cohen group if S is a -group) is defined to be the inverse limit HS = lim pn H n S, where p n = d 0 HnS : H n S H n 1 S. (Note. By the identity on faces, d i HnS = d 0 HnS for each 0 i n.) Lemma 1 (Theorem 1.1.5). Let X and Z be path-connected spaces. Suppose that X is a co-h-space or Z is an H-space. Then [J n+1 (X), Z] = H n K(X, Z) and [J(X), Z] = HK(X, Z).

6 INTRODUCTION 3 In other words, under the above assumptions, the set of homotopy classes [J(X), Z] can be determined by the sets [X n+1, Z] together with face operations and so simplicial techniques can be applied for studying [J(X), Z]. For instance, the Moore-Postnikov system (Theorem 1.2.7) gives a bi- -group resolution of K(X, ΩY ). In the universal cases, the Moore-Postnikov system coincides with the descending central series by Proposition By applying the H-construction to simplicial groups, we have the following surprising results: Theorem 2 (Theorem 1.2.2). Let G = {G n } n 0 be a reduced simplicial group, that is, G 0 = {1}. Then 1) p 2k+1 : H 2k+1 G H 2k G is an epimorphism for each k 0. 2) The image of p 2k : H 2k G H 2k 1 G is a normal subgroup of H 2k 1 G with cokernel isomorphic to π 2k 1 (G) for each k 1. It was conjectured by Fred Cohen that HG is a progroup for any simplicial group G, that is H n G H n 1 G is always onto. Our answer is that HG is a progroup if and only if G has trivial odd dimensional homotopy groups. Moreover the cokernels of H n G H n 1 G are given by the odd dimensional homotopy groups of G. This gives systematic method for obtaining all odd dimensional homotopy groups and killing off all even dimensional homotopy groups. When a -group G admits a bi- -structure, HG is always a progroup according to Proposition Moreover Theorem gives the Taylor series for decomposing elements in the Cohen group HG of a bi- -group G in terms of other elements. This result is a reformulation of the classical distributivity law [1, 5, 8] in the case of K(X, ΩY ) and has applications to the exponent problem. As an example, Example gives the bi- -group structure on pure braids and so the Taylor series gives certain canonical decomposition of certain type of braids. For generalizing the results in [9, 10, 11], we investigate the skeleton filtration of bi- -groups that can be regarded as the categorical interpretation of the skeleton filtration of CW -complexes or the dual version of the Moore-Postnikov systems. Roughly speaking a skeleton of a bi- group G is obtained by taking the lower dimensional groups from G (including faces and cofaces) and freeing up higher dimensional groups to form a bi- -group. It admits the universal property described as in Theorem This gives another bi- -group resolution of K(X, ΩY ). In particular, the 0-skeleton gives the universal group for [ΩΣX, ΩΣX] for X running over all path-connected spaces. According to the remarks to Corollary 1.3.9, this is the progroup built up by the Brunnian braids, where a braid is called Brunnian if it becomes trivial by removing any one of its strings. When X runs over all pathconnected p-local spaces, the p-local version of the universal group is canonically obtained. Moreover various local type of the universal group can be obtained by allowing X to run over certain class of spaces such as localization with respect to a homology theory or a class of maps. According to Proposition 1.3.7, the 0-skeleton is obtained by self free products of a given group with certain canonical faces and cofaces. In addition to the 0-skeleton, it should be pointed out that the higher skeleton of K(X, ΩY ) is also important for individual spaces X and Y. For instance, the divisibility of certain elements occurs in the 1-skeleton of K(S n, ΩY ) if the Whitehead square ω n+1 is divisible by 2. It is a long-standing open problem whether the Whitehead square ω 2 k 1 is divisible by 2. Another example is the bi- -group K(S 1, ΩY ).

7 4 INTRODUCTION As a sequence of groups, this is the bi- -group obtained from [S 1 S 1, ΩY ] with its Cohen group given by [ΩS 2, ΩY ] that is isomorphic to n=1 π n (ΩY ) as a set. The skeleton filtration on K(S 1, ΩY ) is obtained by taking the lower homotopy groups of ΩY and so it is related to the Postnikov system of ΩY. The commutator relations in [9, 10, 11] can be generalized by considering group homomorphisms φ: G [X, ΩY ]. A group homomorphism φ: G [X, ΩY ] is called to have weak LS-category less than k if the composite X k X (k) [[φ(g1),φ(g2)],...,φ(g k)] ΩY is null homotopic for any g 1, g 2,..., g k G, where X (k) is the k-fold self-smash product of X, k is the reduced diagonal and [[φ(g 1 ), φ(g 2 )],..., φ(g k )] is the iterated Samelson product of the maps φ(g 1 ),..., φ(g k ). For instance, any group homomorphism φ: G [X, ΩY ] has weak LS-category less than k if X has weak LS-category less than k (that is, the reduced diagonal k : X X (k) is null homotopic). By using weak LS-category filtration, the universal group admits a resolution by adding certain systematic type of commutator relations. According to Theorem , as the special case that k = 1, the resulting quotient group extends the main results in [9, 10, 11]. Roughly speaking, assuming that X is a path-connected co-h-space with a given subgroup G in [X, ΩY ], then there is a progroup with a group homomorphism to [ΩΣX, ΩY ] that is built up by Lie(n) with coefficients in self tensor products of G, where Lie(n) will be described below. (See Section 1.4 for details.) The group Lie(n) is constructed as follows. Let V be a free Z-module of rank n with a basis {x 1, x 2,..., x n }. The module Lie(n) is defined to be the submodule of V n spanned by the iterated commutators [[x σ(1), x σ(2) ],..., x σ(n) ] for σ S n, where [x, y] = xy yx. (Note. For n = 1, Lie(1) is defined to Z.) The symmetric group S n acts on Lie(n) by permuting the letters {x 1,..., x n }. It is well known that Lie(n) is a free Z-module of rank (n 1)!, see for instance [7, 35]. For any abelian R, write Lie R (n) for Lie(n) Z R. Note that Lie R (n) is a module over the symmetric group algebra R(S n ) if R is a ring. The applications of the symmetric group module Lie R (n) can be found in [39, 40]. Roughly speaking the problem on functorial decompositions of the ΩΣX is equivalent to the problem for determining the maximal projective submodule of Lie R (n) over the symmetric group. Our results described below give further connections between the homological properties of Lie R (n) over the symmetric group and the possible obstructions to the classical exponent problem. In [9], Cohen constructed a progroup H built by Lie(n), that is, there is a tower of group epimorphisms H n H n 1 H 1 = Z such that Ker(H n H n 1 ) = Lie(n) and the group H is given by the inverse limit H = lim H n. n Similarly there is a progroup H R built up by Lie(n) R using our terminology of bi- -groups. It was then proved in [9] that the group H is a subgroup of (homotopy) self natural transformations of the functor ΩΣ 2, that is, H admits a functorially faithful representation e X : H [ΩΣ 2 X, ΩΣ 2 X]. Moreover

8 INTRODUCTION 5 there is a commutative diagram H e X [ΩΣ 2 X, ΩΣ 2 X] H Z/pr Ωf [ΩΣ 2 X, ΩY ] for any map f : Σ 2 X Y of order p r in [Σ 2 X, Y ]. Following the lines in [9] the progroup H Z (p) (H Zp ), built up by Lie(n) over p-local integers Z (p) (p-complete integers Z p ), is a subgroup of (homotopy) self natural transformations of ΩΣ 2 on p-local (p-complete) spaces. It was discovered in [39, 40] that the progroup H R is isomorphic to the group of functorial self coalgebra maps of tensor algebras. More precisely, let R be a given ground ring and let T (V ) be the tensor algebra generated by a projective module V with the canonical Hopf algebra structure by saying V primitive. Consider T : V T (V ) as a functor from projective R-modules to pointed cocommutative coalgebras. Then the set of natural (coalgebra) self transformations of T forms a group under the the convolution product. Let coalg R,graded (T, T ) and coalg R (T, T ) denote the groups of functorial self coalgebra maps of tensor algebras over graded modules and ungraded modules, respectively. Write [ΩΣ 2, ΩΣ 2 ] for the group of homotopy natural transformations of the functor ΩΣ 2. (Note. Following Quillen s ideas, for avoiding set theoretical troubles, ΩΣ 2 is regarded as a functor from CW - complexes with cells indexed by a fixed (large) infinite set to pointed spaces.) Let H R be the progroup constructed in [9, 10, 11] that is built up by Lie(n) with coefficients in R. By using Geometric Realization Theorem [39, Theorem 1.3], the composite H R e [ΩΣ 2, ΩΣ 2 ] H coalg R,graded (T, T ) coalg R (T, T ) = H R is an isomorphism of groups for R = Z, Z (p), Z p, where H is the homology functor with coefficients in R. As a consequence, it follows that H R occurs as a splitting subgroup of [ΩΣ 2, ΩΣ 2 ]. Let T (V ) T (V ) be the quotient module of T (V ) T (V ) by T i (V ) R R R T i (V ) for i > 0. Let ψ be the reduced comultiplication given by the composite ψ : T (V ) ψ T (V ) T (V ) T (V ) T (V ). Then ψ is a functorial coalgebra map and so it induces a group homomorphism ψ : coalg R (T T, T ) coalg R (T, T ) = H R with cokernel denoted by R R. Note that the explicit formula for the ψ is given by the shuffles. The group R R can be regarded as the quotient of H R by certain shuffle type relations. Our next result gives a relation between R R and the group homomorphism Ω: [ΩΣ 2, ΩΣ 2 ] [Ω 2 Σ 2, Ω 2 Σ 2 ] [f] [Ωf]. Let Lie R (n) denote the dual module of Lie R (n). Theorem 3. Let R be a commutative ring with identity. Then there is a quotient group R R of H R = coalg R (T, T ) with the following properties.

9 6 INTRODUCTION 1) R R is an abelian group. Moreover R R is a ring with the multiplication induced by the composition operation on coalg R (T, T ). 2) For any ring homomorphism φ: R R, there is an induced ring homomorphism R(φ): R R R R for changing the group rings, that is, R defines a functor from commutative rings with identity to rings with identity. 3) There is a morphism of rings θ : R R Hom R(Sn)(Lie R (n), Lie R (n)). n=1 If R is a field of characteristic 0, then θ is an isomorphism. 4) There is a tower of epimorphisms of rings such that R R = lim n R R n. R R n R R n 1 R R 1 = R 5) Let I R n denote the kernel of R R n R R n 1. Then there is an exact sequence 0 Ext R(Sn)(Lie R (n), Lie R (n) ) I R n θ Hom R(Sn)(Lie R (n), Lie R (n)) H 0 R(Sn) (LieR (n) ; Lie R (n) ) 0 for each n. 6) If R = Z (p) (or Z p ), there is a commutative diagram of semirings H R R R θ Hom R(Sn)(Lie R (n), Lie R (n)) e [ΩΣ 2, ΩΣ 2 ] e n=1 Ω [Ω 2 Σ 2, Ω 2 Σ 2 Ω k 2 ] [Ω k Σ 2, Ω k Σ 2 ] for each k 2, where Ω k Σ 2 is regarded as a functor from p-local (or p- complete) spaces to pointed spaces. Moreover if f : Σ 2 X Y is of order p r in [Σ 2 X, Y ], then there is a commutative diagram R R e X [Ω 2 Σ 2 X, Ω 2 Σ 2 X] R Z/pr Ω 2 f [Ω 2 Σ 2 X, Ω 2 Y ]. n=1 Remark. The map θ : H R Hom R(Sn)(Lie(n), Lie(n)) is obtained by restricting coalgebra maps to primitives. If R = Z/p r, then, for each t r, the convolution p t -th power of tensor algebras (that corresponds to the p-th power map of the loop suspensions, also to p t id in R R ) lies in Ker(θ) that gives obstructions in the extension group of Lie(n) by part (5). So roughly speaking the (possible) obstructions to the powers of Ω 2 f are displayed in the extension groups

10 INTRODUCTION 7 Ext Z/p r (S n)(lie Z/pr (n), Lie Z/pr (n) ). So far it is not clear whether the representation e: R R [Ω 2 Σ 2, Ω 2 Σ 2 ] is faithful or not for R = Z (p) although by Part (3) of Theorem 3 this is faithful when R = Q. By changing the ground ring Z (p) to be Q, R Z (p) R Q has the kernel that are detected by the extension groups of Lie(n) and so the possible kernel of the representation e in R Z (p) n level is a torsion group. This seems to suggest that there are certain geometric version of the Bocksteins. This project is related to the classical exponent problem and so we give some historical remarks on the exponent problem. Let p be a prime integer and let G be an abelian group. Write Tor p (G) for the p-torsion component of G. The power p r is called an exponent of G if p r Tor p (G) = 0. Let X be a space. The integer p r is called an exponent of π (X) if p r Tor p (π n (X)) = 0 for all n 2. If X is a simply connected CW complex of finite type, then each π n (X) has a bounded exponent which depends on n in general. For instance π (S 2 S 2 ) does not have a bounded exponent. On the other hand, one can check that the homotopy groups of the mapping space from a simply connected finite torsion space to a space has a bounded exponent. It was first known by James [22] that π (S 2n+1 ) has an exponent bounded by 2 2n. The improvements given in [8, 38] state that the exponent of π (S 2n+1 ) is bounded by 2 2n [n/2], where [a] is the maximal integer a. In the cases where p > 2, Toda [44] showed that π (S 2n+1 ) has an exponent bounded by p 2n. Selick [37] then showed that π (S 3 ) has an exponent bounded by p for p > 2. Later Cohen- Moore-Neisendorfer [12] proved that π (S 2n+1 ) has an exponent bounded by p n for p > 2. This is the best exponent because a theorem of Gray [18] gives that there are Z/p n -summands in π (S 2n+1 ) for p > 2. The Barratt conjecture states that if f : Σ 2 X Y is of order p r in the group [Σ 2 X, Y ], then p r+1 Im(f : π (Σ 2 X) π (Y )) = 0; in particular, if the identity map id Σ2 X is of order p r in [Σ 2 X, Σ 2 X], then p r+1 π (Σ 2 X) = 0. It was known by Neisendorfer [31] that the Barratt conjecture holds for (the identity map of) the Moore spaces P n (p r ) with p > 2, but it remains open in general. Cohen proposed a strong form of the Barratt conjecture that if f : Σ 2 X Y is of order p r in the group [Σ 2 X, Y ], then Ω 2 f : Ω 2 Σ 2 X Ω 2 Y has an order bounded by p r+1 in the group [Ω 2 Σ 2 X, Ω 2 Y ]. Note that if this statement is true, then the Barratt conjecture holds for f. Following Cohen s ideas there are essentially two steps for the study of this problem. Firstly study the group [ΩΣ 2 X, ΩY ] for decomposing the powers of [Ωf] as a product of other type of maps, and secondly investigate the group homomorphism Ω: [ΩΣ 2 X, ΩY ] [Ω 2 Σ 2 X, Ω 2 Y ] for hoping to kill out the obstructions to the powers of [Ωf] after looping. The first combinatorial study on the group [ΩΣ 2 X, ΩY ] was given by Fred Cohen [9, 10, 11], where the group H was introduced. Our results give more systematic investigation on the group [ΩΣX, ΩY ]. Roughly speaking, we can start with an arbitrary subgroup of [X, ΩY ] for constructing a universal group for [ΩΣX, ΩY ], rather than starting with a cyclic subgroup generated by a homotopy class [f] [X, ΩY ]. This generalization provides a possible way for collecting more information for studying [ΩΣX, ΩY ]. Moreover our terminology of bi- -groups provides a useful tool for connecting the universal group and the individual group [ΩΣX, ΩY ] by adding possible relations and possible new elements. The examples in [48] show that it is important to make connections between the universal group

11 8 INTRODUCTION and the individual group [ΩΣX, ΩY ], where some special properties of the mod 2 Moore spaces are used. For studying the homomorphism Ω: [ΩΣ 2 X, ΩY ] [Ω 2 Σ 2 X, Ω 2 Y ], Theorem 3 is essentially obtained by considering the reduced diagonal : ΩΣ 2 X (ΩΣ 2 X) (ΩΣ 2 X) and using the well-known result that Ω is null homotopic. The identification of H with functorial coalgebra self maps of tensor algebras gives a useful information for computational purpose using shuffles. It seems that these are the first type of canonical relations by applying the cobar resolution to ΩΣ 2 X. The obstruction groups I R n can be also described as the top cohomology of the Artin braid group B n with coefficients in Lie(n) by the following theorem. For an R(S n )-module M, write M[ 1] for the module M with signed S n -action, that is, M[ 1] = M R[ 1]. Theorem 4. Let I R n be given in Theorem 3 and let B n act on Lie R (n) via the canonical quotient B n S n. Then there is an isomorphism I R n = H n 1 (B n ; Lie R (n)[ 1]) = H n 1 (B n ; H n 1 (P n ; R)) where P n = Ker(B n S n ) is the Artin pure braid group with the canonical B n - action on H (P n ; R). This theorem gives some connections with the complexity of algorithms studied by Smale [42] and other people [15, 16, 17]. The article is organized as follows. In Section 1.1, we go over the James construction, where Lemma 1 is Theorem Bi- -groups are introduced in Section 1.2, where some basic properties such as the Taylor series (Theorem 1.2.4) and the Postnikov system (Theorem 1.2.7) are given. Theorem 2 is Theorem In Section 1.3, a categorical skeleton filtration of bi- -groups is given. The generalized Cohen construction is given in Section 1.4. In Section 2.1, we investigate functors to coalgebras. The geometric realizations are discussed in Section 2.2 and the proof of Theorem 3 is given in Section 2.3. The proof of Theorem 4 is given in Section 2.4. The author is indebted to Professors Jon Berrick, Fred Cohen, Bill Dwyer, John Harper and Paul Selick for fruitful conversations on this project. These conversations defined what was most important in this paper. The author would like to thank the referee for his/her important comments and helpful suggestions. The author also would like to thank many of our colleagues for their encouragement and suggestions on this project.

12 CHAPTER 1 Maps from Loop Suspensions to Loop Spaces 1.1. The James Construction In this section, we go over the James construction. Some of the results in this section may be well-known. Let X be a pointed space and let J(X) be the James construction with the James filtration {J n (X)}. For a subspace A of X, write (X k A) n for the subspace of X n consisting of points (x 1, x 2,..., x n ) X n with at least k coordinates lie in A. We simply write (X A) n for (X 1 A) n. Note that X n /(X ) n = X (n) and (X n 1 ) n = n j=1 X. Let q n : X n J n (X) be the canonical quotient map. Then there is a natural commutative diagram n (X n 1 ) n = X (X 1 ) n X n (1.1.1) J 1 (X) = X j=1 J n 1 (X) q n J n (X) for any (pointed) space X. In particular, there is a commutative diagram of cofibre sequences (CX X) n (CX) n (ΣX) (n) n J n 1 (ΣX) J n (ΣX) (ΣX) (n), where CX = X [0, 1] is the cone of X. The map n : (CX X) n J n 1 (ΣX) is called the higher Whitehead product, studied in [32, 33, 34]. Since (CX) n is contractible, we have the following: Proposition Let X be any pointed space. There is a (functorial) cofibre sequence Thus the cofibre sequence is principal. (CX X) n Σ n 1 X (n) n Jn 1 (ΣX) Jn (ΣX). J n 1 (ΣX) J n (ΣX) (ΣX) (n) Corollary Let X be any pointed space. There is a (functorial) coaction of (ΣX) (n) on J n (ΣX), µ : J n (ΣX) J n (ΣX) (ΣX) (n), such that the 9

13 10 1. MAPS FROM LOOP SUSPENSIONS TO LOOP SPACES diagram of cofibre sequences commutes functorially. J n 1 (ΣX) J n (ΣX) (ΣX) (n) µ J n 1 (ΣX) J n (ΣX) (ΣX) (n) (ΣX) (n) (ΣX) (n) From the classical suspension splitting theorem [21], q n : [J n (X), Y ] [X n, Y ] is a monomorphism if Y is a loop space. This result can be generalized as follows: Proposition Let X and Y be pointed spaces. Then the kernel of the function q n : [J n (X), Y ] [X n, Y ] is trivial, that is, if f : J n (X) Y such that f q n is null homotopic, then so is f. Proof. The proof is given by induction on n. Clearly the statement holds for n = 1. Suppose that the statement holds for n 1 with n 2. Let f : J n (X) Y be any map such that f q n is null homotopic. From the commutative diagram X n X n 1 q n 1 q n J n 1 (X) J n (X), the map f restricted to J n 1 (X) is null homotopic and hence there is a map g : X (n) Y such that f is homotopic to the composite J n (X) p n X (n) = J n (X)/J n 1 (X) Consider the commutative diagram of cofibre sequence J n 1 (X) J n (X) (X ) n X n q n p n X (n) p n q n X (n) g Y. ΣJ n 1 (X) Σ(X ) n. Since g p n q n = f q n is null homotopic by the assumption, there exists a map h: Σ(X ) n Y such that g h. From the suspension splitting, the inclusion Σ(X ) n ΣX n admits a retraction ΣX n Σ(X ) n. Thus the boundary map : X (n) Σ(X ) n is null homotopic. It follows the map g is null homotopic and so is f. The induction is finished and hence the result. Corollary Let X and Y be pointed spaces. Suppose that X is a suspension or Y is a loop space. Then the function is one-to-one. q n : [J n (X), Y ] [X n, Y ]

14 1.1. THE JAMES CONSTRUCTION 11 Proof. 1 If Y is a loop space, the assertion follows from Proposition because q n : [J n (X), Y ] [X n, Y ] is a group homomorphism. If X is a suspension, the proof is given by induction on n. Clearly the statement holds for n = 1. Suppose that the statement holds for n 1 with n 2. Let f, g : J n (X) Y be any maps such that f q n g q n. Since f Jn 1(X) q n 1 = f q n X n 1 g q n X n 1 = g Jn 1(X) q n 1, f Jn 1(X) g Jn 1(X) by induction. By Corollary 1.1.2, there is a map h: X (n) Y such that the homotopy class [g] = [f] [h] in [J n (X), Y ]. It follows that [f q n ] = [g q n ] = [f q n ] [h] in [X n, Y ]. It is routine to check that the action of [X (n), Y ] on [X n, Y ] is free. Thus [h] = 1 and so [f] = [g]. The induction is finished and hence the result. Let X and Y be pointed spaces. The map d i : X n X n+1, (z 0, z 1,..., z n 1 ) (z 0,..., z i 1,, z i,..., z n 1 ) induces a function (group homomorphism if Y is a loop space) d i = d i : [X n+1, Y ] [X n, Y ] for 0 i n. Recall that a sequence of sets (groups) S = {S n } n 0 is called a -set if there are functions (group homomorphisms) d i : S n S n 1, 0 i n, such that the identity (1.1.2) d j d i = d i d j+1 holds for all j i. Let K(X, Y ) denote the sequence of sets {[X n+1, Y ]} n 0 with faces d i = d i. It is straight forward to check that the above equality holds for d i and so K(X, Y ) is a -set ( -group if Y is a loop space). Let S be a -set. The Cohen set (Cohen group if S is a -group) H n S is defined by H n S = {x S n d 0 x = d 1 x = = d n x}, namely, H n S is the equalizer of the faces d i for 0 i n. Observe that the face d 0 : S n S n 1 induces p n : H n S H n 1 S such that S n d i S n 1 H n S p n H n 1 S commutes for each 0 i n. The total Cohen set (total Cohen group if S is a -group) HS is defined to be the inverse limit HS = lim pn H n S. For the -set K(X, Y ), the quotient map q n : X n J n (X) induces functions q n : [J n (X), Y ] Hn 1 K(X, Y ), 1 The proof was omitted in the original manuscript. The author would like to thank referee s suggestion to include a proof to this corollary. This proof is also outlined by the referee.

15 12 1. MAPS FROM LOOP SUSPENSIONS TO LOOP SPACES [J(X), Y ] lim[j n (X), Y ] lim n q n HK(X, Y ). n Theorem Let X and Y be path-connected spaces. Suppose that X is a co-h-space or Y is an H-space. Then 1) the function q n : [J n (X), Y ] H n 1 K(X, Y ) is one-to-one and onto; 2) the function [J(X), Y ] HK(X, Y ) is one-to-one and onto. Proof of Theorem in the case that Y is an H-space. If Y is a loop space, a proof of assertion (1) can be found in [48, Lemma 2.9] and then assertion (2) follows easily from the suspension splitting theorem of J(X). Now assume that Y is a path-connected H-space. Then the inclusion j : Y ΩΣY admits a retraction r : ΩΣY Y. The assertions follows from the commutative diagram [J n (X), Y ] Hn 1 K(X, Y ) j j = [J n (X), ΩΣY ] Hn 1 K(X, ΩΣY ) r [J n (X), Y ] r Hn 1 K(X, Y ) with r j = id. We are going to prove the theorem in the case that X is a co-h-space. We need some lemmas. For each sequence I = (i 1, i 2,..., i k ) with 0 i 1 < < i k n 1, let d I : X k (X n k ) n denote the composite d i k d i1, where d i : X s X s+1 is defined above. In other words, d I (x 1, x 2... x k ) (X n k ) n is a point whose (i s + 1) st coordinate is x s, 1 s k, and the rest coordinates are the basepoint. (For instance, for k = 1 and n = 2, d 0 (x 1 ) = (x 1, ) and d 1 (x 1 ) = (, x 1 ).) Let I run over all sequences (i 1,..., i k ) with 0 i 1 < < i k n 1. The maps d I : X k (X n k ) n define a map θk n : X k (X n k ) n. 0 i 1<i 2< <i k n 1 The proof of the following lemma is similar to that of Proposition Lemma Let X and Y be pointed spaces. Suppose that n k 1. Then the kernel of the function θk n : [(X n k ) n, Y ] [X k, Y ] = [ X k, Y ] 0 i 1<i 2< <i k n 1 0 i 1<i 2< <i k n 1 is trivial.

16 1.1. THE JAMES CONSTRUCTION 13 From the commutative diagram of cofibre sequences (ΣX n k 1 ) n (ΣX n k ) n 0 i 1<i 2< <i k n 1 0 i 1<i 2< <i k n 1 (ΣX ) k (CX X) k we have the following: θ n k 0 i 1<i 2< <i k n 1 0 i 1<i 2< <i k n 1 (ΣX) k (CX) k 0 i 1<i 2< <i k n 1 0 i 1<i 2< <i k n 1 0 i 1<i 2< <i k n 1 Lemma Let X and Y be pointed spaces. Suppose that n k 1. Then the function θk n : [(ΣX n k ) n, Y ] [(ΣX) k, Y ] is one-to-one. 0 i 1<i 2< <i k n 1 Let s: J n (X) J n (X) = J n (X) I (J n 1 (X) I) (J n (X) I) be the inclusion of subspaces of X I, where I = [0, 1] with I = {0, 1} and the half-smash A B = (A B)/( B). Lemma Suppose that X is a suspension. Then s : [(J n 1 (X) I) (J n (X) I), Y ] [Jn (X) J n (X), Y ] is one-to-one for any space Y. Proof. Let X = ΣZ and let A denote (J n 1 (X) I) (J n (X) I). Consider the commutative diagram J n (X) = J n (X) 0 =========== J n (X) s 0 A φ j Jn (X) I X (n) S 1 ((CZ Z) n I) ((CZ) n I) (CZ) n I X (n) S 1, where the rows are cofibrations. The commutative diagram implies that the composite J n (X) (((CZ Z) n I) ((CZ) n I)) s0 φ A A fold A (ΣX) (k) (ΣX) (k) (ΣX) (k),

17 14 1. MAPS FROM LOOP SUSPENSIONS TO LOOP SPACES is a homotopy equivalence. Now consider the homotopy commutative diagram J n (X) 0 J n (X) I p n Jn (X) 1 = J n (X) X (n) s 0 s 1 J n (X) 0 s 0 r A ((CZ Z) n I) ((CZ) n I) X (n), where p n is the quotient map, r is a retraction of the map φ and the rows except the last terms are splitting cofibre sequences. Since the boundary map X (n) ΣJ n 1 (X) is null homotopic, the function p n : [X (n), Y ] [J n (X), Y ] is one-to-one and so, from the above commutative diagram, the function (s 0 s 1 ) : [A, Y ] [Jn (X) I, Y ] = [J n (X) J n (X), Y ] is one-to-one. This finishes the proof. Proof of Theorem in the case that X is a co-h-space. Since any path-connected co-h-space is a retract of a suspension, we may assume that X is a suspension. (1). By Corollary 1.1.4, the function q n : [J n (X), Y ] Hn 1 K(X, Y ) is one-to-one. We show that q n is onto by induction on n. Clearly q 1 is onto. Suppose that q k is onto for k < n with n 2. Let f : Xn Y be a map such that the homotopy class [f] lies in H n 1 K(X, Y ), that is, f d 0 f d 1 f d n 1 : X n 1 Y. Since [f d 0 ] = d 0 [f] H n 2 K(X, Y ), there is a map g : J n 1 (X) Y such that f d 0 g q n 1 by induction. From the pushout diagram (X ) n cofibration X n j q n J n 1 (X) q n Jn (X), it suffices to show that f j g q n : (X ) n Y because, if so, the map f will factors through q n up to homotopy. Let n θn 1 n : X n 1 (X ) n [ n ] be the map defined in Lemma Then, in the set X n 1, Y = s=1 s=1 n [X n 1, Y ], [g q n θ n n 1] = ([g q n 1 ],..., [g q n 1 ]) = ([f d 0 ], [f d 1 ],..., [f d n 1 ]) = [f j θ n n 1] s=1

18 1.1. THE JAMES CONSTRUCTION 15 because d i [f] = d 0 [f] for all i. By Lemma 1.1.7, θ n n 1 is one-to-one and so g q n f j. Thus assertion (1) follows. (2). It suffices to show that the function [J(X), Y ] limn [J n (X), Y ] is one-to-one. Let f, g : J(X) Y be maps such that f Jn(X) g Jn(X) : J n (X) Y for all n. We are going to construct homotopies F n : J n (X) I Y between f Jn(X) and g Jn(X) such that F n+1 Jn(X) I = F n by induction on n starting with a homotopy F 1 : f X g X : X I Y. Suppose that F n 1 : J n 1 (X) I Y is defined such that F n 1 (x, 0) = f(x) and F n 1 (x, 1) = g(x) for x J n 1 (X). Let G = f F n 1 g : (J n (X) 0) (J n 1 (X) I) (J n (X) 1) Y and let F : J n (X) I Y be a homotopy between f and g. Observe that G(x) = F (x) = (f g)(x) for x J n (X) I = J n (X) J n (X). By Lemma 1.1.8, the map F restricted to the subspace (J n (X) I) (J n 1 (X) I) is homotopic to G. Since the inclusion (J n (X) I) (J n 1 (X) I) J n (X) I is a cofibration, the map G admits an extension F n : J n (X) I Y. The induction is completed and so the map This finishes the proof of Theo- F = F n : J(X) n is a well-defined homotopy between f and g. rem The following universal property is useful. Y Proposition Let f : X ΩY be a pointed map. Then there exists an unique, up to homotopy, H-map Jf : J(X) ΩY such that Jf X = f. Proof. Clearly any pointed map f : X ΩY admits an extension of H-map from J(X) ΩY. It suffices to show the uniqueness. Let f, f : J(X) ΩY be two H-maps such that f X f X f : X ΩY. Consider the homomorphisms of monoids f, f : [X n, J(X)] [X n, ΩY ]. Let x i [X n, J(X)] represented by the composite X n π i X J(X), where π i is the i th coordinate projection. Then f (x i ) = f (x i ) = [f π i ] for 1 i n. Thus f (x 1 x 2 x n ) = f (x 1 x 2 x n ) [X n, ΩY ] for each n. Observe that the element x 1 x 2 x n is represented by the composite X n q n Jn (X) J(X) by the canonical multiplication of J(X). We obtain that f Jn(X) q n f Jn(X) q n for each n. By Corollary 1.1.4, f Jn(X) f Jn(X) : J n (X) ΩY

19 16 1. MAPS FROM LOOP SUSPENSIONS TO LOOP SPACES for each n. From Theorem 1.1.5, f f : J(X) ΩY and hence the result Bi- -groups In this section, some properties of bi- -groups are investigated. A sequence of sets S = {S n } n 0 is called a bi- -set if there are faces d j : S n S n 1 and cofaces d j : S n 1 S n for 0 j n such that the following identities hold: (1.2.1) d j d i = d i d j+1 for j i, (1.2.2) d j d i = d i+1 d j for j i, d i 1 d j for j < i, (1.2.3) d j d i = id for j = i, d i d j 1 for j > i, where d 0 x = for x S 0. In other words, S is a - and co- -set such that relation (1.2.3) holds. Moreover a sequence of groups G is called a bi- -group if G is a bi- -set such that all faces and cofaces are group homomorphisms. A weak bi- -group means a bi- -set G = {G n } n 0 such that each G n is a group and all faces are group homomorphisms, namely the cofaces are not required to be group homomorphisms. Recall that the Moore complex NG = {N n G} n 0 of a -group G is defined by n N n G = Ker(d i : G n G n 1 ) with d 0 : N n G N n 1 G. The groups of the Moore cycles ZG and the Moore boundaries are defined by Z n G = Ker(d i : G n G n 1 ) and B n G = d 0 (N n+1 G), i 0 respectively. The Moore homotopy group π n (G) is defined to be the coset π n (G) = H n (NG, d 0 ) = Z n G/B n G. (Note. B n G need not be normal in Z n G for general -group. Some properties of -groups have been studied in [3].) The homotopy groups of bi- -groups seem less interesting as one can easily check that the homotopy groups of a bi- -group are all trivial. But the cycles in bi- -groups are interesting, for instance, if G is the bi- -group given by G n = [T n+1, ΩY ] for n 0, then Z n 1 G = π n (ΩY ) = π n+1 (Y ), where T n is the n-fold Cartesian product of S 1. The Cohen group HG of a weak bi- -group G is always a progroup by the following proposition. Proposition Let G be a weak bi- -group. Then the map p n : H n G H n 1 G is an epimorphism with the kernel Z n G for each n. Proof. The proof follows from the lines in the proof of [48, Lemma 2.10]. It should be pointed out that if G is a simplicial group, the map p n : H n G H n 1 G is not onto in general and its cokernel is related to the homotopy groups π (G). As a comparison between simplicial groups and bi- -groups, we give the following statement on simplicial groups which seems interesting independently. Theorem Let G = {G n } n 0 be a reduced simplicial group, that is, G 0 = {1}. Then 1) p 2k+1 : H 2k+1 G H 2k G is an epimorphism for each k 0.

20 1.2. BI- -GROUPS 17 2) The image of p 2k : H 2k G H 2k 1 G is a normal subgroup of H 2k 1 G with cokernel isomorphic to π 2k 1 (G) for each k 1. Proof. We refer the terminology on simplicial sets to [14, 24]. Let [n] denote the standard n-simplex with the only non-degenerate element σ n [n] n. Let E[n] be the quotient simplicial set of [n] by requiring d i σ n d 0 σ n for each 0 i n. Let q n : [n] E[n] be the quotient simplicial map and let σ n = q n (σ n ) E[n] n. Then d i σ n = d 0 σ n for all 0 i n. By simplicial relations, d i1 d i2 d ik σ n = d j1 d j2 d jk σ n for all sequences (i 1, i 2,..., i k ) and (j 1, j 2,..., j k ). In other words, E[n] has one-cell in each dimension up to n. Let f diσ n : [n 1] [n] be the simplicial map by sending σ n 1 to d i σ n. Then there is a unique simplicial injection j n : E[n 1] E[n] such that the diagram [n 1] f d iσ n [n] q n 1 q n E[n 1] jn E[n] commutes for each 0 i n. In other words, E[n 1] can be regarded as the simplicial subset of E[n] by taking cells up to dimension n 1 under the map j n. Observe that H (E[n]; Z) = H (C), where C is a chain complex given by C i = 0 for i > n and C i = Z for 0 i n with the faces k = k ( 1) j d j (E[n]) = j=0 k ( 1) j : C k C k 1 for 1 k n. It follows that E[2k] is contractible for each k 0 and the pinch map j=0 E[2k + 1] E[2k + 1]/E[2k] = S 2k+1 is a homotopy equivalence after geometric realization. Let Hom(X, Y ) denote the set of simplicial maps. Given any simplicial set S = {S n } n 0. Recall that, for each x S n, there is a unique simplicial map f x : [n] S, called representing map of x, such that f x (σ n ) = x. Furthermore, the function f : S n Hom( [n], S) x fx is one-to-one and onto (group isomorphism if S is a simplicial group), see [14] for details. Observe that for x H n G, the representing map f x : [n] G factors

21 18 1. MAPS FROM LOOP SUSPENSIONS TO LOOP SPACES through the quotient E[n]. This gives the commutative diagram G n f Hom( [n], G) = q n H n G f Hom(E[n], G). = Now we start to prove the assertions: (1). Let x H 2k G and let f x : E[2k] G be the representing map of x. Since E[2k] is contractible and G is fibrant, there is an extension map g : E[2k + 1] G such that g E[2k] = f x : E[2k] G. Let y = g( σ 2k+1 ). Then d i y = g(d i σ 2k+1 ) = g( σ 2k ) = f x ( σ 2k ) = x for 0 i 2k + 1. This proves that H 2k+1 G H 2k G is an epimorphism. (2). Let x H 2k 1 G and let f x : E[2k 1] G be the representing map. Consider the cofibration E[2k 1] E[2k]. Since E[2k] is contractible, the map f x extends to E[2k] if and only if f x is null homotopic. Thus K = Im(H 2k G H 2k 1 G) = {x H 2k 1 G f x }. Let x, y H 2k 1 G be represented by f x and g x respectively. The the commutator [x, y] = x 1 y 1 xy is represented by the composite E[2k 1] E[2k 1] E[2k 1] fx gx G G [, ] G, since E[2k 1] S 2k 1 with 2k 1 1, the reduced diagonal : E[2k 1] E[2k 1] E[2k 1] is null homotopic. It follows that the commutator subgroup [H 2k 1 G, H 2k 1 G] is contained in K. In particular, K is a normal subgroup of H 2k 1 G with H 2k 1 G/K = Hom(E[2k 1], G)/{f : E[2k 1] G f } = [E[2k 1], G] = π 2k 1 (G). Remark We gives some remarks to Theorem 1.2.2: 1) The reduced condition of G is used for technical reason that any simplicial map f : X G is automatically a pointed simplicial map and any homotopy F : X I G is automatically a pointed homotopy. Possibly this condition can be removed. 2) Let E = n=0 E[n]. Then HG = Hom(E, G). The group HG depends on simplicial group G rather than its homotopy type. But the cokernel of H 2k G H 2k 1 G only depends on the homotopy type of G as it is just the homotopy group. 3) This result somehow reveals that there is a possible way to kill off all even homotopy groups (or odd homotopy groups) by making certain constructions on simplicial groups. In our construction, HG is a tower of groups rather than the traditional simplicial groups.

22 1.2. BI- -GROUPS 19 Let G be a weak bi- -group. For any integers k n, the James-Hopf operation H k,n : G k G n is defined by H k,k = id and, for n > k, H k,n (x) = d i n k d i n k 1 d i1 (x) 0 i 1<i 2< <i n k n with lexicographic order from right. If x Z k G, then it is straightforward to check that d i H k,n (x) = H k,n 1 (x) for 0 i n and k < n, and so H k,n (x) H n G for any n k with p n H k,n (x) = H k,n 1 (x). This determines a unique element H k (x) HG that projects to H k,n (x) in H n G for each n k (in case x Z k G). Note. When G = {[X n+1, ΩY ]} n 0, the subgroup Z k G H k G is the group of the homotopy classes represented by the composites J k+1 (X) pinch X (k+1) f ΩY for any map f : X (k+1) ΩY. Given any element x f Z k G H k G represented by J k+1 (X) pinch X (k+1) f ΩY, then the element Hk (x f ) HG = [J(X), ΩY ] is represented by the composite J(X) H k+1 J(X (k+1) ) J(f) ΩY, where the map H k+1 is the James-Hopf invariants and J(f) is the H-map such that J(f) X (k+1) = f. If ΩY = J(X (k+1) ) and f is given by the inclusion X (k+1) J(X (k+1) ), then H k (x f ) is the homotopy class [H k+1 ] [J(X), J(X (k+1) )]. In general, the above composite can be rewritten as a formula H k (x f ) = Jf ([H k+1 ]) for any map f : X (k+1) ΩY. This describes the operation H k as the evaluations on the James-Hopf invariants in the case when G = {[X n+1, ΩY ]} n 0. The James-Hopf operation H k has the following property. Theorem (Taylor Series). Let G be a weak bi- -group. Then, for any α HG, there exists an unique element δ k (α) Z k G for k 0 such that Proof. The functions n φ n : Z k G α = H k (δ k (α)). k=0 Q n k=0 H k,n n k=0 k=0 H n G µ Hn G induces a unique function φ = lim n φ n : Z k G k=0 HG. It suffices to show by induction that φ n is an isomorphism for each n 0. Note that φ 0 : Z 0 G = G 0 G 0 is the identity map. Suppose that φ n 1 is an isomorphism

23 20 1. MAPS FROM LOOP SUSPENSIONS TO LOOP SPACES with n > 0. From the commutative diagram of Z n G-sets Z n G n n 1 Z k G Z k G k=0 k=0 φ n = φ n 1 Z n G Hn G Hn 1 G, where Z n G acts on H n G and n k=0 Z kg as subgroups and the short exactness of the bottom row is from Proposition 1.2.1, the function φ n is an isomorphism and hence the result. Remark We give some remarks to Theorem 1.2.4: 1) From the proof, the function δ n : HG Z k G can be defined recursively as follows: δ 0 (α) = α G0 = α H0G, and δ n (α) = α HnG H n 1,n (δ n 1 (α)) 1 H n 2,n (δ n 2 (α)) 1 H 0,n (δ 0 (α)) 1. The function δ n provides a recursive way for constructing cycles from H n G. The simplicial analogues might be the algorithms for constructing higher dimensional cycles in simplicial groups such as Samelson products in [14]. 2) Let φ: G G be a morphism of weak bi- -groups. Then φ(α) = H k (φ(δ k (α))) k=0 for any α HG. That is, the Taylor series is functorial. 3) Let G = {[X n+1, ΩY ]} n 0 and let α HG = [J(X), ΩY ]. Then H 0 (δ 0 (α)) = J(α): J(X) ΩY is the H-map induced by δ 0 (α) = α X : X ΩY. The higher terms H k (δ k (α)) = J(δ k (α)) H k+1 with δ k (α) being recursively given in (1). The Taylor series gives a decomposition formula of α and so it becomes an infinite summation in [ΩJ(X), Ω 2 Y ] after looping. 4) Assume that X is path-connected. Let G = {[X n+1, ΩΣX]} n 0 and let α = q HG = [J(X), ΩΣX] be the q th power map of J(X) ΩΣX. Then the Taylor series gives a decomposition, called the distributivity law [8, Theorem 2.2.1], ( ) q Ω[q] J(δ k (q)) H k+1 k=1 in [ΩΣX, ΩΣX], where [q]: ΣX ΣX is the map of degree q and δ k (q): X (k+1) J(X) is a product of certain (k + 1)-fold iterated Samelson products. Suppose that [p r ]: ΣX ΣX is null homotopic. Then p r = H k (δ k (p r )) k=1

24 1.2. BI- -GROUPS 21 and so the Barratt conjecture is equivalent to that: p H k (δ k (p r )) : π (ΩΣX) π (ΩΣX (k+1) ) k=1 k=1 π (ΩΣX) is a zero map if [p r ]: ΣX ΣX is null homotopic. Thus the self maps H k (δ k (p r )): ΩΣX H k+1 ΩΣX (k+1) Jδ k(p r ) ΩΣX can be regarded as the obstructions to the Barratt conjecture. 5) Given an element α [J(X), ΩY ], the explicit computation for the element δ k (α): X (k+1) ΩY seems hard although it can be recursively defined by (1). However in the special cases when X is a sphere localized at 2, the 3-fold Whitehead products are trivial and so the Taylor series only has two terms as described in [48, Example 2.11]. Similarly if ΩY is a two-stage Postnikov system, the computations becomes relatively much easier which might help to understand the secondary operations on the cohomology of J(X). 6) It should be also pointed out that, by using the functorial property of Taylor series, one may get information on the classical distributivity law by considering bi- morphisms into (or from) the bi- -group {[X n+1, ΩY ]} n 0. For instance, for spaces X with the property that X = ΣX and the degree map [p r ]: X X is null homotopic, the distributivity law for [J(X), J(X)] lifts to the Cohen group introduced in [11]. The Cohen group for these spaces admits a faithful representation to the group of functorial self-coalgebra maps of tensor algebras over the ground ring R = Z/p r according to Theorem By considering convolution powers p r+t of tensor algebras over the ground ring R = Z/p r, one obtains that the convolution power p r+t is trivial restricted to the sub-coalgebra of tensor algebras having tensor-length p t+1 1. It follows that the power map p r+t of J(X) is null homotopic restricted to J p t+1 1(X). (See Example ) Lemma Let G be a weak bi- -group. Then G satisfies the following strong fibrant condition Let x 0, x 1,, x n be elements in G n 1 such that d i x j = d j 1 x i for i j. Then there exists an element w G n such that d i w = x i for 0 i n. Proof. We define the elements w i G n for 0 i n recursively by putting w 0 = d 0 x 0 and w i = w i 1 (d i d i w i 1 ) 1 d i x i for i > 0. We show by induction that d j w i = x j for j i. The assertion will follow by taking w = w n. Clearly d 0 w 0 = y 0. Suppose that d j w i 1 = x j for j i 1 with i > 0. Then d i w i = x i and, for j < i, d j w i = d j w i 1 (d j d i d i w i 1 ) 1 d j d i x i = d j w i 1 (d i 1 d i 1 d j w i 1 ) 1 d i 1 d j x i = x j (d i 1 d i 1 x j ) 1 d i 1 d i 1 x j = x j. The induction is finished and hence the result. Let G be a -group. Let R n G be the sequence of groups defined by 1 0 q n (R n G) q = {x G q d i1 d i2 d iq n x = 1 for all (i 1, i 2,..., i q n )} q > n

25 22 1. MAPS FROM LOOP SUSPENSIONS TO LOOP SPACES with faces d i (R n G) = d i (G). Observe that each (R n G) q is a normal subgroup of G q. Let (P n G) q = G q /(R n G) q for each q 0. The sequence of -groups (1.2.4) G Pn G Pn 1 G P0 G is called the Moore Postnikov system of G. Let q n : G P n G be the quotient map. The following theorem tells that the progroup HG can be obtained by considering the Moore-Postnikov system. Theorem Let G be a weak bi- -group. Then 1) each R n G is a weak bi- -subgroup and so the Moore-Postnikov system is a tower of weak bi- -groups. 2) q n : Z k G Z k P n G is an isomorphism for k n. 3) Z k P n G = 1 for k > n and so HP n G = H n P n G. 4) q n : H n G H n P n G is an isomorphism. Proof. Assertion (1) is a routine exercise. Assertion (2) follows immediately from that G k = (P n G) k for k n. Assertion (4) follows from assertions (2) and (3). We prove assertion (3). Let x Z k P n G with k > n. Since q n : G P n G is onto, there exists y G k such that q n (y) = x. Since d j x = 1 for all j, the elements d 0 y, d 1 y,, d k y has matching faces in (R n G) k 1 and so, by Lemma 1.2.6, there exists an element w (R n G) k such that d j w = d j y for 0 j k. It follows that yw 1 Z k G (R n G) k with x = q n (yw 1 ) = 1 and hence the result. Example Let M be a manifold. Consider the configuration space F (M, n + 1) = {(z 0, z 1,..., z n ) z i M, z i z j for i j}. Let B(M, n + 1) = F (M, n + 1)/S n+1, where S n+1 acts on F (M, n + 1) by permuting coordinates. According to [3, Proposition 4.2.1], the sequence of fundamental groups B(M) π1 = {π 1 (B(M, n + 1))} n 0 is a crossed -group, that is, the face functions d i : π 1 (B(M, n + 1)) π 1 (B(M, n)) satisfies the rule that d i (ββ ) = d i (β)d β(i) (β ), where the action of π 1 (B(M + 1)) on {0, 1,..., n} is induced from the covering map F (M, n + 1) B(M, n + 1). The i-th face d i is essentially induced from the coordinate projection F (M, n + 1) F (M, n) by deleting (i + 1) st coordinate. (See [3] for details.) Suppose that M is a path-connected compact manifold with non-empty boundary M. Let a be a point in M. Then the map F (M, n) F (M a, n) F (M, n + 1), (z 0, z 1,..., z n 1 ) (z 0,..., z i 1, a, z i,..., z n 1 ), induces a group homomorphism d i : π 1 (B(M, n)) π 1 (B(M, n + 1)) for 0 i n. The sequence of groups B(M) π1 with faces and cofaces is then a bi- set with the property that B(M) π1 is a co- group under cofaces and a crossed -group under faces. In particular, the sequence of pure braid groups {π 1 (F (M, n + 1))} n 0 is a bi- -group. If M = D 2, then B(M) π1 = {B n+1 } n 0 is the sequence of the classical braid groups B n+1. According to [3], the face operation d i on the braids is obtained by deleting the (i + 1) st string. From the above construction, the coface operation d i