Combinatorial operad actions on cochains Clemens Berger Benoit Fresse 14/8/2001 Abstract A classical E-infinity operad is formed by the bar construction of th* *e symmetric groups. Such an operad has been introduced by M. Barratt and P. Eccles in the con* *text of simplicial sets in order to have an analogue of the Milnor FK-construction for infinite l* *oop-spaces. The purpose of this article is to prove that the associative algebra structure on the* * normalized cochain complex of a simplicial set extends to the structure of an algebra over the Barra* *tt-Eccles operad. We also prove that differential graded algebras over the Barratt-Eccles oper* *ad form a closed model category. Similar results hold for the normalized Hochschild cochain comp* *lex of an associative algebra. More precisely, the Hochschild cochain complex is acted on by a* * suboperad of the Barratt-Eccles operad which is equivalent to the classical little square * *operad. Introduction This article follows the work of M. Mandell which compares the algebras ove* *r an E1 - operad to the homotopy category of simplicial sets (cf. [18]). We consider the * *normalized cochain complex N*(X) associated to a simplicial set X. Our main purpose is to * *provide the complex N*(X) with the structure of an algebra over a suitable E1 -operad. * * The results of M. Mandell imply that any such structure determines the homotopy typ* *e of the simplicial set X (under reasonable finiteness and completeness assumptions on X* *). There is a classical E1 -operad which is denoted by the letter E in this ar* *ticle and whose component E(r), r = 0, 1, . .,.is given by the normalized bar constructio* *n of the symmetric group r. The operad structure is provided by an explicit composition* * product: E(r) E(s1) . . .E(sr) -! E(s1 + . .+.sr). The degree 0 component of E(r) is the regular representation of the symmetric g* *roup r and is identified with the module generated by the multilinear monomials in * *r non- commutative variables. Hence, the degree 0 elements represent operations assoc* *iated to the structure of an associative algebra. In this article, the differential grad* *ed operad E is called the Barratt-Eccles operad, because the associated simplicial operad W is* * introduced by M. Barratt and P. Eccles for the study of infinite loop spaces (cf. [1]). We obtain the following result: Theorem For any simplicial set X, the normalized cochain complex N*(X) is equipped * *with the structure of an algebra over the Barratt-Eccles operad E. In fact, the existence of such an algebra structure is given by a Theorem o* *f Justin Smith (cf. [28]). Our purpose is to give an explicit construction for it. This * *is achieved by the Theorems 1.3.2 and 2.2.7 in the article. As a corollary, we obtain a simple* * formula for 1 the evaluation product E(r) N*(Sn) r - ! N*(Sn) on the cochain complex of the spheres Sn (cf. Theorem 3.2.4 and Proposition 3.2* *.5). The Barratt-Eccles operad comes equipped with a diagonal E(r) -! E(r) E(r* *) be- cause it is the chain operad associated to a simplicial operad. This structure * *has interesting applications which make the Barratt-Eccles operad suitable for calculations in * *homotopy theory. For instance, we prove that differential graded algebras over the Barr* *att-Eccles operad form a closed model category. (It is not clear that this property is sa* *tisfied for algebras over a general E1 -operad.) In fact, we have a more general result. To* * be pre- cise, a classical construction replaces an operad Q by the operad P = E Q suc* *h that P(r) = E(r) Q(r). The operad P is a *-projective resolution of Q. The Barrat* *t-Eccles operad P = E corresponds to the case Q = C of the commutative operad. The follo* *wing Theorem is proved in Section 3.1: Theorem Let Q be an operad. We let P = E Q be the standard *-projective resolut* *ion of Q. The P-algebras are equipped with a closed model category structure for w* *hich the weak equivalences (respectively, the fibrations) are the algebra morphisms whic* *h are weak equivalences (respectively, fibrations) in the category of dg-modules. The simplicial Barratt-Eccles operad has also a filtration F1W F2W . . .FnW . . .W by simplicial suboperads FnW whose topological realization |FnW| is equivalent * *as a topo- logical operad to the classical operad of little n-cubes (cf. C. Berger [3], T.* * Kashiwabara [15], J. Smith [27]). We have an induced filtration on the associated differen* *tial graded operad F1E F2E . . .FnE . . .E. The operad F1E is identified with the associative operad. The operad F2E is th* *e chain operad on a simplicial operad equivalent to the little square operad. The next * *result implies that this operad F2E gives a solution to a conjecture of Deligne: Theorem The normalized Hochschild cochain complex of an associative algebra is equi* *pped with the structure of an algebra over the operad F2E. The pivot of our constructions is formed by a quotient operad X of the Barr* *att-Eccles operad E which we call the surjection operad. This operad is also introduced (w* *ith different sign conventions) by J. McClure and J. Smith in their work on the Deligne conje* *cture (cf. [21], [22]) to which we refer in this article. Our results follow from the nex* *t Theorem. In fact, the surjection operad X acts naturally on the cochain complex associat* *ed to a simplicial set. In this context, the idea of the surjection operad goes back to* * D. Benson (cf. [2]) and to the original construction of the reduced square operations by * *N. Steenrod (cf. [29]). The introduction of the surjection operad in the context of the Del* *igne conjecture 2 is due to J. McClure and J. Smith. In fact, the surjection operad X is endowed* * with a filtration F1X F2X . . .FnX . . .X by suboperads FnX such that F1X is identified with the associative operad. Furt* *hermore, one observes that the operad F2X acts naturally on the Hochschild cochain compl* *ex of an associative algebra (cf. [21], [22]). Theorem The surjection operad X is a quotient of the Barratt-Eccles operad E. More * *precisely, there is a surjective morphism of filtered operads T R: E - ! X such that F1 T * *R: F1E - ! F1X is the identity isomorphism of the associative operad. We call the quotient morphism T R : E -! X the table reduction morphism, * *by reference to its construction. J. McClure and J. Smith have proved that the surjection operad FnX is equiv* *alent to the chain-operad of the little n-cubes. (As a consequence, the operad F2X is al* *so a solution to the Deligne conjecture.) We prove more precisely that the table reduction m* *orphism induces a weak equivalence Fn T R: FnE - ! FnX . But, since the operads FnX are* * not directly associated to a simplicial operad, it is not clear that some of our co* *nstructions are available in the context of algebras over the surjection operad. This moti* *vates the introduction of the Barratt-Eccles operad and of the table reduction morphism. We refer to Section 0 for the conventions on operads which are adopted thro* *ughout the article. The detailed definition of the Barratt-Eccles operad is given in Secti* *on 1.1. The surjection operad is introduced in Section 1.2. Section 1.3 is devoted to the c* *onstruction of the table reduction morphism. The properties of the morphism are proved in Sect* *ions 1.4 and 1.5. (These two Sections may be skipped in a first reading.) The little cub* *e filtration on the operads and the applications to the Deligne conjecture are explained in * *Section 1.6. The construction of the algebra structure on the normalized cochain complex of * *a simplicial set is achieved in Section 2. The applications of our constructions to homotopi* *cal algebra are explained in Section 3. Contents 0. Conventions 1. The operads 1.1. The Barratt-Eccles operad 1.2. The surjection operad 1.3. The table reduction morphism 1.4. The operad morphism properties 1.5. On the composition products 1.6. The little cube filtrations on operads 2. The interval cut operations on chains 2.1. On the Eilenberg-Zilber operad 2.2. The interval cut operations in the Eilenberg-Zilber operad 3 3. On closed model structures 3.1. The closed model structure 3.2. On spheres, cones and suspensions 3.3. Some proofs Acknowledgements: The morphisms from the Barratt-Eccles operad to the surjection operad and t* *o the Eilenberg-Zilber operad have been constructed (modulo the signs) by the second * *author and lectured on at the fall-school öL la" in Malaga in November 2000. The secon* *d author would like to thank the organizers and the participants of the school who provi* *ded the first motivation for this work. We thank Jim McClure and Jeff Smith for their interes* *t on our work which, at some places, should complement their results on the Deligne conj* *ecture. In fact, the first author realized that the Barratt-Eccles operad provides a so* *lution to the Deligne conjecture after reading the announcement of the article [22]. 4 x0. Conventions 0.1. On permutations We let r, r 2 N, denote the permutation groups. The element 1r 2 r is the identical permutation 1r = (1, 2, . .,.r). In general, a permutation oe 2 r is* * specified by the sequence (oe(1), oe(2), . .,.oe(r)) formed by its values. The permutation oe 2 r determines a block permutation oe*(s1, . .,.sr) 2 * *s1+...+sr (for s1, . .,.sr 2 N). In the sequence (oe(1), oe(2), . .,.oe(r)), we replace * *the occurence of k = 1, . .,.r by the sequence 1, 2, . .,.sk together with a shift of s1 + s2 + * *. .+.sk-1. For instance, if oe is the transposition oe = (2, 1), then oe*(p, q) is the blo* *ck transposition oe*(p, q) = (p + 1, . .,.p + q, 1, . .,.p). 0.2. On the symmetric monoidal category of dg-modules Let us fix a ground ring F. The category of F-modules is denoted by Mod F* *. The category of differential graded modules is denoted by dgMod F. If V is a differential graded module (or, for short, a dg-module), then V * * has either an upper V = V * or a lower V = V* graduation. The equivalence between lower* * and upper graduations is given by the classical rule. To be explicit, the component* * of upper degree d is equivalent to the component of lower degree -d of the dg-module. T* *hus: V *= V-*. Let us mention that differential degrees are assumed to range over th* *e integers without any restriction. In general, the differential of a dg-module is denoted* * by the letter ffi : V * -! V *+1. The category of differential graded modules is equipped with its classical * *tensor prod- uct : dgMod Fx dgMod F- ! dg Mod F. If U, V 2 dgMod F, then U V 2 dgMod Fis the dg-module which has the module M (U V )n = Ud V n-d d2Z in degree n. The differential of U V is given by the derivation rule: ffi(u v) = ffi(u) v + (-1)du ffi(v), for all homogeneous elements u 2 Ud and v 2 V e. The symmetry operator ø : U * *V -! V U follows the rule of signs. If u 2 Ud and v 2 V e, then we have: ø(u v) = (-1)dev u. In general, a permutation of homogeneous symbols, which have d and e as degrees* *, produces the sign (-1)de. For instance, in the derivation rule, the permutation of the * *differential ffi, which has degree 1, with the element u, homogeneous of degree d, produces * *the sign (-1)d. A morphism f : U - ! V is homogeneous of upper degree d if it raises upper * *degrees by d and lowers lower degrees by d. The differential of this morphism ffi(f) : * *U - ! V is given by the commutator of f with the differentials of U and V : ffi(f) = ffi . f - (-1)df . ffi. 5 The dg-module formed by the homogeneous morphisms f : U -! V is denoted by Hom *F(U, V ). Thus, we have explicitly: Y Y Hom dF(U, V ) = Hom F(U*, V *+d) = Hom F(U*+d, V*). *2Z d2Z This dg-module Hom *F(U, V ) is an internal hom in the category of dg-modules. 0.3. The symmetric monoidal categories In fact, we are concerned with the following classical symmetric monoidal c* *ategories. 1) The category of F-modules Mod F together with the classical tensor product: : Mod F x Mod F -! Mod F. 2) The category of dg-modules dgMod Ftogether with the tensor product of dg-m* *odules: : dgMod Fx dgMod F- ! dg Mod F, whose definition is recalled in the Paragraph above. 3) The category of sets Set together with the cartesian product: x : Setx Set -! Set. 4) The category of simplicial sets S together with the cartesian product of si* *mplicial sets: x : S x S - ! S. Let us also introduce the symmetric monoidal category of (differential grad* *ed) coal- gebras. Classically, a coalgebra is a (differential graded) module X 2 Mod F to* *gether with an associative diagonal : X - ! X X. If X and Y are coalgebras, then the te* *nsor product X Y 2 Mod F is equipped with a canonical diagonal given by the compos* *ite X Y --! X X Y Y (2-3)-!X Y X Y. It follows that the coalgebras form a symmetric monoidal category. 0.4. On normalized chains and cochains If X is a simplicial set, then C*(X) denotes the dg-module whose degree d c* *om- ponent Cd(X) is the free F-module generated by the set Xd of the d-dimensional * *sim- plices inPX. The differential of x 2 Xd in C*(X) is given by the classical for* *mula: ffi(x) = di=0(-1)idi(x). The normalized chain complex N*(X) is the quotient * *of the dg-module C*(X) by the degeneracies: Nd(X) = Cd(X)=s0Cd-1(X) + . .+.sd-1Cd-1(X). We consider also the dual cochain complexes C*(X) = Hom F(C*(X), F) and N*(X) = Hom F(N*(X), F). 6 Let us recall that the normalized chain complex X 7! N*(X) is a (symmetri* *c) monoidal functor from the category of simplicial sets to the category of dg-mod* *ules. More precisely, the Eilenberg-Zilber morphism provides a natural equivalence EZ : N*(X) N*(Y ) -~-!N*(X x Y ) which is associative and commutes with the symmetry isomorphism. The dg-module N*(X) is also equipped with a natural diagonal. This diagona* *l is given by the classical Alexander-Whitney formula. Since, furthermore, the diag* *onal is compatible with the Eilenberg-Zilber morphism, we conclude that the normalized * *chain complex X 7! N*(X) is a (symmetric) monoidal functor from the category of simpl* *icial sets to the category of dg-coalgebras. We refer to the classical textbook of S. Mac Lane (cf. [17, Chapter VIII])* * for the properties of the Eilenberg-Zilber equivalence. 0.5. On the notion of an operad The main structures involved in our article are operads in the symmetric mo* *noidal category of dg-modules. But, for simplicity, we recall the definition of an op* *erad in the category of F-modules. In fact, the notion of an operad makes sense in any sym* *metric monoidal category. For an introduction to the subject, we refer to the works of* * E. Getzler and J. Jones (cf. [9]), V. Ginzburg and M. Kapranov (cf. [10]) and to the works* * of P. May (cf. [20]). A *-module M is a sequence M = M(r), r 2 N, where M(r) is a representation* * of the permutation group r. A morphism of *-modules f : M -! N is a sequence of morphisms of representations f : M(r) -! N(r). An operad P is a *-module equipped with a composition product P(r) P(s1) . . .P(sr) -! P(s1 + . .+.sr), defined for r 1, s1, . .,.sr 0, and which statisfies some natural equivaria* *nce and associativity properties. A morphism of operads f : P -! Q is a morphism of* * *- modules which commutes with the operad composition product. In general, an element p 2 P(r) represents a (multilinear) operation in r v* *ariables x1, . .,.xr. Thus, we may write p = p(x1, . .,.xr). The action of a permutati* *on oe 2 r on p 2 P(r) is given by the permutation of the variables. Explicitly, we have * *oe . p = p(xoe(1), . .,.xoe(r)). The composition product of p 2 P(r) and q1 2 P(s1), . .* *,.qr 2 P(sr) is denoted by p(q1, . .,.qr) 2 P(s1 + . .+.sr). In fact, this operation is giv* *en by the substitution of the variables x1, . .,.xr by the operations q1, . .,.qr. We assume that the composition product of an operad has a unit 1 2 P(1). T* *his unit represents the identical operation 1(x1) = x1. In this context, there are * *also partial composition products: Ok : P(r) P(s) -! P(r + s - 1), for k = 1, . .,.r. If p 2 P(r) and q 2 P(s), then p Ok q = p(1, . .,.1, q, 1, * *. .,.1). (The operation q is set at the kth entry of p.) 7 0.6. On algebras over an operad The operad P has an associated category of algebras denoted by AlgP . The * *set of the P-algebra morphisms from A 2 AlgP to B 2 AlgP is denoted by Hom P (A, B). T* *o fix a definition, a P-algebra is an F-module A equipped with an evaluation product P(r) A r - ! A, equivariant and defined for any r 0. This evaluation product has to be associ* *ative with respect to the operad composition product and unital with respect to the operad* * unit. The image of the elements p 2 P(r) and a1, . .,.ar 2 A under the evaluation pro* *duct is denoted by p(a1, . .,.ar) 2 A. The equivariance relation reads (oe . p)(a1,* * . .,.ar) = p(aoe(1), . .,.aoe(r)). A morphism of P-algebras f : A - ! B is a morphism of F* *-modules which commutes with the evaluation product. The free P-algebra generated by an* * F- module V is denoted by P(V ). 0.7. The morphism operad If V is an F-module, then the morphism operad associated to V is the operad* * such that Hom V (r) = Hom F(V r, V ). The composition product of the morphism opera* *d is just given by the composition of multilinear maps. If f : V r -! V , then oe . f : * *V r -! V is defined by f(v1, . .,.vr) = f(voe(1), . .,.voe(r)). Observe that V is an alg* *ebra over Hom V . This algebra structure is universal. If P is an operad, then a structure of a P* *-algebra on V is equivalent to an operad morphism P - ! Hom V (cf. [10]). 0.8. On operads in symmetric monoidal categories We have mentioned that the notion of an operad makes sense in any symmetric monoidal category. One has just to replace the classical tensor product by any* * other tensor product in the definitions. For instance, there are operads in the categ* *ory of sets and in the category of simplicial sets. In this case, the tensor product is rep* *laced by the cartesian product. The symmetric group is always assumed to operate on the left. 0.9. On the normalized chain complex associated to a simplicial operad If O is a simplicial operad, then the associated normalized chain complexes* * P(r) = N*(O(r)) form an operad in the category of dg-coalgebras, because the normalize* *d chain complex functor N*(-) is symmetric monoidal as reminded in the Paragraph 0.4. E* *xplic- itly, the dg-module N*(O(r)) is equipped with the Alexander-Whitney diagonal N*(O(r)) -! N*(O(r)) N*(O(r)). The composition product is defined as the composite of the composition product * *of O with the Eilenberg-Zilber equivalence: N*(O(r)) N*(O(s1)) . . .N*(O(sr)) -! N*(O(r) x O(s1) x . .x.O(sr)) -! N*(O(s1 + . .+.sr)). 8 0.10. On Hopf operads Operads in the monoidal category of coalgebras are also known as Hopf opera* *ds (cf. [9]). Explicitly, a Hopf operad is an operad in the category of dg-modules P to* *gether with a diagonal on each component P(r) -! P(r) P(r), r 2 N. The composition product P(r) P(s1) . . .P(sr) -! P(s1 + . .+.sr) has to commute with the diagonal. If P is an Hopf operad, then a P-algebra denotes a (differential graded) mo* *dule to- gether with a P-algebra structure. In fact, a P-algebra in the symmetric monoid* *al category of coalgebras is called a Hopf-P-algebras. (We should not consider such structu* *res.) Here is the property of Hopf operads which interests us. If A, B 2 AlgP are P-algebr* *as, then the tensor product A B 2 Mod F is equipped with a natural P-algebra structure* *. The evaluation product of A B is given by the composite of the evaluation product* *s of A and B together with the diagonal of the operad: P(r) (A B) r--! (P(r) P(r)) (A B) r --! (P(r) A r) (P(r) B r ) --! A B. P Explicitly, let ipi(1) pi(2)2 P(r) P(r) be thePdiagonal of an operation p 2 P* *(r). Then, in A B, we have p(a1 b1, . .,.ar br) = ipi(1)(a1, . .,.ar) pi(2)(b1, * *. .,.br). 0.11. The permutation operad The permutation groups r, r 2 N, form an operad in the category of sets. * *This operad is called the permutation operad. The composition product r x s1x . .x. sr -! s1+...+sr is characterized by the relation 1r(1s1, . .,.1sr) = 1s1+...+sr. Thus, if u 2 r and v1 2 s1, . .,.vr 2 sr, then, in s1+...+sr, we have the r* *elation u(v1, . .,.vr)= v1 . . .vr . u*(s1, . .,.sr) = u*(s1, . .,.sr) . vu(1) . . .vu(r). In fact, the composed permutation u(v1, . .,.vr) 2 s1+...+srcan be obtained by* * the fol- lowing explicit substitution process. In the sequence (u(1), u(2), . .,.u(r)), * *we replace the occurence of k = 1, . .,.r by the sequence (vk(1), vk(2), . .,.vk(sk)) together* * with the ap- propriate shift on the indices. Precisely, we increase the index vk(j) by s1+s2* *+. .+.sk-1. The process is the same for a partial composition product u Ok v = u(1, . .,.1, v, 1, . .,.1) 2 r+s-1. In this case, we have just to replace the occurence of k by the sequence (v(1)+* *k -1, v(2)+ k - 1, . .,.v(s) + k - 1) and to increase by s the elements u(i) such that u(i)* * > k. For instance, if u = (3, 2, 1) and v = (1, 3, 2), then we obtain u O2 v = (5, 2, 4,* * 3, 1). 9 0.12. The associative and the commutative operads The algebras over the permutation operad are just the associative monoids. * *There is also an operad in the category of sets, whose algebras are the associative and * *commutative monoids. We consider the associated operads in the category of F-modules. The associative operad A denotes the operad in the category of F-modules wh* *ose algebras are the associative algebras. The module A(r) is generated by the mul* *tilinear monomials in r non-commutative variables. Explicitly, we have: M A(r) = (i F . xi1. .x.ir 1,...,ir) where (i1, . .,.ir) ranges over the permutations of (1, . .,.r). The commutative operad C is the operad in the category of F-modules whose a* *lge- bras are the associative and commutative algebras. The module C(r) is generated* * by the multilinear monomials in r commutative variables. Thus, we have C(r) = F . x1 . .x.r and C(r) is the trivial representation of the symmetric group r. 10 x1. The operads 1.1. The Barratt-Eccles operad 1.1.1. Summary In general, an E1 -operad E is a *-projective resolution of the commutativ* *e operad C. By definition, an operad is *-projective if it is projective as a *-module* *. (In fact, a *-module M is projective if each component M(r) is a projective representation* * of the symmetric group.) Furthermore, an operad E is a resolution of the commutative o* *perad C, if it is endowed with an augmentation morphism E - ! C which induces an isomorp* *hism from the homology of E to the the commutative operad C. There is a canonical E1 -operad E given by the bar construction on the symm* *etric groups. The aim of this section is to recall the definition and to make explici* *t the structures of this operad. Let us mention that the degree 0 component of E is equal to the* * associative operad. In fact, we have a factorization A --!E -~-!C of the operad morphism from the associative to the commutative operad. In addit* *ion, the operad E is equipped with a diagonal E(r) -! E(r) E(r) and forms a Hopf operad. In the article, the dg-operad E is referred to as the Barratt-Eccles operad* *. In fact, the dg-operad E is the normalized chain complex on a simplicial operad W introduced* * by M. Barratt and P. Eccles for the study of infinite loop spaces (cf. [1]). 1.1.2. The dg-module The dg-module E(r) is the normalized homogeneous bar complex associated to * *the symmetric group r. To be more explicit, the module E(r)d is the F-module gene* *rated by the d + 1-tuples (w0, . .,.wd), where wi 2 r, i = 0, 1, . .,.d. If some co* *nsecutive permutations wi and wi+1 are equal, then, in E(r)d, we have (w0, . .,.wd) = 0. * * The differential of E(r) is given by the classical formula: Xd ffi(w0, . .,.wr) = (-1)i(w0, . .,.cwi, . .,.wd). i=0 The permutations oe 2 r act diagonaly on E(r)*: oe . (w0, . .,.wd) = (oe . w0, . .,.oe . wd). The quasi-isomorphism E(r) -~-!C(r) is the classical augmentation E(r)0 --!* *F from the homogeneous bar construction to the trivial representation. The degree 0 co* *mponent of E(r) is clearly the regular representation of the symmetric group r. Theref* *ore, we have a canonical morphism of dg-modules A(r) -! E(r). 11 It is possible to specify an element (w0, . .,.wd) 2 E(r) by a table with d* * + 1 rows indexed by 0, . .,.d. The row i consists of the sequence (wi(1), . .,.wi(r)) an* *d determines the permutation wi. For instance, the table: fifi1, 2 fifi2, 1 fifi1, 2 fi2, 1 represents the element (w0, w1, w2, w3) 2 E(2)3 such that w1 = w3 = (2, 1) is t* *he transpo- sition of 2 and w0 = w2 = (1, 2) is the identity permutation of 2. 1.1.3. The composition product The composition product of the operad E is induced by the composition produ* *ct of permutations. More precisely, if u = (u0, . .,.ud) 2 E(r)d and v = (v0, . .,.v* *e) 2 E(s)e, then, in E(r + s - 1)d+e, we have the identity X u Ok v = (x (ux0 Ok vy0, . .,.uxd+eOk vyd+e). *,y*) The sum ranges over the set of paths in an d x e-diagram such as: 0____0__//____1//_2_______________________3d=4_______* *________________ __0 ___1 |2 ___ ___ ____ ____ | ____ ____ 1______________fflffl|_______________________________* *_______________//_______________________ ___ ____ ____3 |4 ____ ____ ____ ____ | ____ 2_____________________fflffl|________________________* *________________________________________________________ ____ _____ _____ |5 _____ ____ _____ _____ | _____ e=3_____________________fflffl|________________________* *__________________67//_ The indices involved in the sum are given by the coordinates of the vertices. I* *n the example above, we have: (x*, y*) = (0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (3, 2), (3, 3), * *(4, 3). The sign associated to a term is determined by a permutation of the horizontal * *and vertical segments of the path. More precisely, one considers the shuffle permutation whi* *ch takes the horizontal segments to the first places and the vertical segments to the la* *st places. The sign is just defined as the signature of the shuffle. In the example, the segme* *nts (0, 1), (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7) are permuted to (0, 1), (1, 2), (3, 4), (6, 7), (2, 3), (4, 5), (5, 6) and, as a result, the associated sign is -1. 12 1.1.4. The diagonal There is also a diagonal : E(r) -! E(r) E(r) which represents the cup-pro* *duct on the cohomology of the symmetric groups. This diagonal is given by the classical* * formula: Xn (w0, . .,.wn) = (w0, . .,.wd) (wd, . .,.wn), d=0 for any (w0, . .,.wn) 2 E(r)d. 1.1.5. On the Barratt-Eccles operad As mentioned in the introduction of the section, the operad E is associated* * to a simplicial operad W (the classical Barratt-Eccles operad). Let us recall the de* *finition of the operad W. If X is a (discrete) set, then W(X) denotes the classical contractible simp* *licial set associated to X, which has W(X)n = X_x_._.x.X-z____" n+1 as a set of n-dimensional simplices. The faces and degeneracies of W(X) are giv* *en by the omission and repetition of components. Explicitly, if (x0, . .,.xn) 2 W(X)n, th* *en we have di(x0, . .,.xn)= (x0, . .,.bxi, . .,.xn), fori = 0, 1, . .,* *.n, and sj(x0, . .,.xn)= (x0, . .,.xj, xj, . .,.xn), forj = 0, 1, .* * .,.n. Observe that X 7! W(X) defines a (symmetric) monoidal functor from the category* * of sets to the category of simplicial sets. We have W(X x Y ) = W(X) x W(Y ). As* * a consequence, if O is an operad in the category of sets, then W(O) is an operad * *in the category of simplicial sets. The Barratt-Eccles operad is formed by the contractible simplicial sets W(r* *) = W( r), r 2 N, associated to the permutation operad. To be explicit, the simpli* *ces (w0, . .,.wn) 2 W( r)n support the diagonal action of the symmetric group. We h* *ave: oe . (w0, . .,.wn) = (oe . w0, . .,.oe . wn), for any oe 2 r. Furthermore, given u = (u0, . .,.un) 2 W( r)n and v = (v0, . * *.,.vn) 2 W( s)n, the composite operation u Ok v 2 W( r+s-1)n is given by the formula: u Ok v = (u0 Ok v0, . .,.un Ok vn). The augmentation maps W( r) -~-!pt define a morphism of simplicial operads to the (set) commutative operad. The identity E(r) = N*(W( r)) is clear. Furthermore, the diagonal E(r) -! E* *(r) E(r) defined in Paragraph 1.1.4 is clearly the Alexander-Whitney diagonal on th* *e normal- ized chain complex N*(W( r)). Our definition of the composition product in the * *operad E follows from a classical explicit representation of the Eilenberg-Zilber equi* *valence: Nd(W( r)) Ne(W( s)) -~-!Nd+e(W( r) x W( s)) (cf. [7]). 13 1.2. The surjection operad The surjection operad X is a quotient of the Barratt-Eccles operad E. The s* *tructure of the surjection operad X is made explicit in this Section. The quotient morp* *hism is introduced in the next section. 1.2.1. The module of surjections The module X (r)d is generated by the non-degenerate surjections u : {1, . * *.,.r+d} -! {1, . .,.r}. The degenerate surjections, whose definition is given below, are e* *quivalent to zero in X (r)d. The non-surjective maps represent also the zero element 0 2 X (* *r)d. In the context of the surjection operad, a map u : {1, . .,.r + d} -! {1, . .,.r} is r* *epresented by a sequence (u(1), . .,.u(r +d)). By definition, a surjection u : {1, . .,.r +d}* * -! {1, . .,.r} is degenerate if the associated sequence contains a repetition (more explicitly* *, if we have u(i) = u(i + 1) for some 0 < i < r + d). The module X (r)d has a canonical r-m* *odule structure. In fact, if oe 2 r, then the surjection oe . u 2 X (r)d is represe* *nted by the sequence (oe(u(1)), . .,.oe(u(r + d))). In the sequel, we say that v = (v(1), . .,.v(s)) is a subsequence of u = (u* *(1), . .,.u(r + d)) if v(1) = u(i1), v(2) = u(i2), . .,.v(s) = u(is), for some 1 i1 < i2 < . * *.<.is r + d. 1.2.2. The table arrangement of a surjection We define the table arrangement of a surjection u 2 X (r)d. In fact, the ta* *ble arrange- ment is specified by the decomposition of u in certain subsequences ui= (u(r0_+_._.+.ri-1+_1)-z________", u(r0_+_._.+.ri-1+_2)-z________", . .* *,.u(r0_+_._.+.ri-1+_ri)-z_________") ui(1) ui(2) * * ui(ri) where r0, r1, . .,.rd 1 and r0 + r1 + . .+.rd = r + d. But, in general, the * *values of the surjection u are just arranged on a table with d + 1 rows indexed by 0, . .* *,.d. The subsequence ui consists of the terms on the row i: fifiu (1), . . .u (r - 1), u (r ), fifiu0(1), . . .u0(r0 - 1), u0(r0), fi . 1 1. 1 1. 1 u(1), . .,.u(r + d) = fifi.. .. .. fifiu (1),. .u. (r - 1), u (r ), fiu d-1 d-1 d-1 d-1 d-1 d(1), . . .ud(rd). Therefore, the subsequence ui is referred to as the line i of the table arrange* *ment of the surjection u. The elements u0(s0), u1(s1), . .,.ud-1(sd-1) which are at the end of a row (except for the last one) are the caesuras of the* * surjection. The caesuras are fixed as the elements of the sequence (u(1), . .,.u(r+d)) which do* * not represent the last occurence of a value k = 1, . .,.r. As an example, the sequence u = (1* *, 3, 2, 4, 2, 1) has the following table arrangement: fifi1, fi3, 2, 1, 3, 2, 1, 4, 2, 1 = fififi1, fi4, 2, 1. 14 In general, if u(i1), . .,.u(in-1), u(in)) are the occurences of k = 1, . .* *,.r, then the element u(in) is the final occurence of k. The elements u(i1), . .,.u(in-1) are* * also referred to as the inner occurences of k. Thus, by definition, the inner occurences are* * caesuras of the surjection. The final occurences are inside a row of the table arrangeme* *nt. These latter elements u0(1), . .,.u0(s0 - 1), . .,.ud-1(1), . .,.ud-1(sd-1 - 1), ud(1), . .,.* *ud(sd), form a permutation. 1.2.3. The complex of surjections In this Paragraph, we equip the module of surjections with a differential f* *fi : X (r)* ! X (r)*-1. The identity ffi2 = 0 follows from results which we prove in the next* * section. By definition, if the surjection u 2 X (r)d is specified by the sequence (u* *(1), . .,.u(r + d)), then the differential ffi(u) 2 X (r)d-1 is the sum: r+dX ffi(u(1), . .,.u(r + d)) = (u(1), . .,.du(i), . .,.u(r + d)). i=1 The signs are determined by the table arrangement. Precisely, the caesuras of t* *he surjection are marked with alterned signs. Otherwise, if u(i) is inner a line, then u(i) i* *s by definition the last occurence of k = u(i) in the sequence. If k occurs only once, then the* * sequence (u(1), . .,.du(i), . .,.u(r + d)) does not represent a surjection and, therefor* *e, vanishes in X (r)d-1. Otherwise, we mark u(i) with the opposite of the sign associated to t* *he previous occurence of k. As an example, for the surjection u = (1, 3, 2, 4, 2, 1) 2 X (4)2, we obtai* *n the signs: fifi1+ , fifi3, 2- , fifi1+ , fi4, 2+ , 1- . Therefore, we have ffi(1, 3, 2, 4, 2, 1) = (3, 2, 4, 2, 1)-(1, 3, 4, 2, 1)+(1, * *3, 2, 4, 1)-(1, 3, 2, 4, 2). 1.2.4. The operadic composition of surjections We equip the module of surjections with a partial composition product X (r)d X (s)e -! X (r + s - 1)d+e. In fact, if u = (u(1), . .,.u(r + d)) 2 X (r)d a* *nd v = (v(1), . .,.v(s + e)) 2 X (s)e, then the product u Ok v 2 X (r + s - 1)d+e is g* *iven by the substitution of the occurences of k in (u(1), . .,.u(r+d)) by elements of (v(1)* *, . .,.v(s+e)). Precisely, assume that k has n occurences in (u(1), . .,.u(r + d)) which ar* *e the terms u(i1), . .,.u(in). In this case, we decompose the sequence (v(1), . .,.v(s + e)* *) in n compo- nents: (v(j0), . .,.v(j1)) (v(j1), . .,.v(j2)) . . .(v(jn-1), . .,.v(jn* *)) where 1 = j0 j1 j2 . . .jr-1 jn = s + e. Then, in (u(1), . .,.u(r + d)* *), we replace the term u(im ) by the sequence (v(jm-1 ), . .,.v(jm )). Furthermore, * *we increase 15 the indices v(j) by k - 1. The indices u(i) such that u(i) < k are fixed. The i* *ndices u(i) such that u(i) > k are increased by s - 1. The sequence which arise from this * *process represents a composed surjection. The composition product uOkv is the sum of the composed surjections togethe* *r with a sign which we specify in the next Paragraph. The sum ranges over all the decomp* *ositions of (v(1), . .,.v(s + e)). As an example, we have: (1, 2, 1, 3) O1 (1, 2, 1) = (1, 3, 1, 2, 1, 4) (1, 2, 3, 2, 1, 4) * *(1, 2, 1, 3, 1, 4). In fact, there are 2 occurences of k = 1 in (1, 2, 1, 3). Therefore, we cut the* * sequence (1, 2, 1) in 2 pieces. There are 3 possibilities (1)(1, 2, 1), (1, 2)(2, 1) and (1, 2, 1)* *(1). Hence, the sub- stitution gives the composed surjections (1, 3, 1, 2, 1, 4), (1, 2, 3, 2, 1, 4)* * and (1, 2, 1, 3, 1, 4). 1.2.5. The signs in the composition of surjections In fact, our substitution process requires some sequence-permutations and t* *hese per- mutations determine a sign which has to appear in the expansion of the composit* *ion product. In more details, the subsitution process works as follows. The sequence (u(* *1), . .,.u(r+ d)) is cut on the occurences of k. Thus, with the notation above, we obtain exp* *licitly: (u(1), . .,.u(i1)) (u(i1), . .,.u(i2)) . . .(u(in-1), . .,.u(in)) (u(in),* * . .,.u(r + d)). Let us define also i0 = 0 and in+1 = r +d. The sequence (v(1), . .,.v(s+e)) is * *decomposed in (v(j0), . .,.v(j1)) (v(j1), . .,.v(j2)) . . .(v(jn-1), . .,.v(jn* *)). The sequence (v(jm-1 ), . .,.v(jm )) is inserted on the position of u(im ). The* *refore, if we move the sequence from the right to the left, then we have to permute this sequ* *ence with the components (u(in), . .,.u(in+1)) (u(in-1), . .,.u(in)) (u(im ), . .,.u(im+1 )* *). The sign is precisely produced by these transpositions according to the permuta* *tion rule in differential graded calculus. Then, we just concatenate the sequences in ord* *er to achieve the subsitution. The sign associated to the permutation depends on a degree which equips the* * com- ponents of a surjection. By definition, the sequence (u(im-1 ), . .,.u(im )) ha* *s degree d0 if it intersects d0+ 1 lines in the table arrangement of the surjection (u(1), . .* *,.u(r + d)). In fact, if dm , m = 1, . .,.n+1, denotes the degree of the sequence (u(im-1 ), . * *.,.u(im )), then the definition implies that (u(im-1 ), . .,.u(im )) intersects the lines: d1 + . .+.dm-1 , d1 + . .+.dm-1 + 1, . .,.d1 + . .+.dm-1 + dm . The degree of the components (v(jm-1 ), . .,.v(jm )) of (v(1), . .,.v(s + e)) a* *re fixed by the same rule. 16 1.2.6. Example As an example, let us determine the signs which occur in the composition pr* *oduct (1, 2, 1, 3) O1 (1, 2, 1) = (1, 3, 1, 2, 1, 4) (1, 2, 3, 2, 1, 4) * *(1, 2, 1, 3, 1, 4) considered above. The sequence (1, 2, 1, 3) has the following table arrangement: fifi 1, 2, 1, 3 = fifi1,2, 1, 3. This sequence is decomposed in (1)(1, 2, 1)(1, 3). The component (1) which is * *contained in the first line has degree 0. The component (1, 2, 1) intersects the lines 1* * and 2 and has degree 1. The component (1, 3) contained in the last line has degree 0. The* * degrees of the components of (1, 2, 1) are determined similarly. Now, we perform the f* *ollowing permutations and concatenations (the subscripts denote the degree of the compon* *ents): (1)0(1, 3, 1)1(1, 4)0(1)0(1, 2, 1)1 7! +(1)0(1)0(1, 3, 1)1(1, 2, 1)1(1, 4)0 7!* * +(1, 3, 1, 2, 1, 4), (1)0(1, 3, 1)1(1, 4)0(1, 2)1(2, 1)0 7! -(1)0(1, 2)1(1, 3, 1)1(2, 1)0(1, 4)0 7!* * -(1, 2, 3, 2, 1, 4), (1)0(1, 3, 1)1(1, 4)0(1, 2, 1)1(1)0 7! -(1)0(1, 2, 1)1(1, 3, 1)1(1)0(1, 4)0 7!* * -(1, 2, 1, 3, 1, 4), from which we deduce the sign associated to each term of the product (1, 2, 1, * *3)O1(1, 2, 1). Explicitly, we obtain finally: (1, 2, 1, 3) O1 (1, 2, 1) = (1, 3, 1, 2, 1, 4) * *- (1, 2, 3, 2, 1, 4) - (1, 2, 1, 3, 1, 4). We claim that our definitions make sense: 1.2.7. Proposition The constructions above provide the graded module of surjections with the s* *tructure of a dg-operad. Explicitly, we prove that the composition product of surjections is compati* *ble with the differential and satistifies the unit, associativity and equivariance propertie* *s of an operad composition product. In fact, we deduce the proposition from results which we p* *rove in the next sections. The surjection operad is also a *-projective resolution of the * *commutative operad (cf. [22]). But, we do not use this result in the article. 1.3. The table reduction morphism 1.3.1. The table reduction morphism We define a morphism T R : E -! X (the table reduction morphism). Let w = (w0, . .,.wd) 2 E(r)d. Given indices r0, . .,.rd 1 such that r0 + . .+.rd = * *r + d, we form a surjection w0= (w0(1), . .,.w0(r + d)) 2 X (r)d from the values of the p* *ermutations w0, . .,.wd. In the sequel, we say that the surjection w0 2 X (r)d arise from * *the simplex w 2 E(r)d by a table reduction process. The image of w = (w0, . .,.wd) in X (r)* * is the sum of these elements X T R(w0, . .,.wd) = (r (w0(1), . .,.w0(r + d)) 0,...,rd) 17 for all choices of r0, . .,.rd 1. Let us mention that there are only positive* * signs in the sum. Fix r0, . .,.rd 1 as above. The associated surjection has a table arrange* *ment fifiw0(1), . . .w0(r - 1), w0(r ), fifiw00(1),. . .w00(0r - 1), w00(0r ), fi . 1 1. 1 1. 1 w0(1), . .,.w0(r + d) = fifi.. .. .. fifiw0 (1),. .w.0 (r - 1), w0 (r ), fiw0 d-1 d-10 d-1 d-1 d-1 d(1), . . .wd(rd), whose rows are formed by the values of the permutations w0, . .,.wd. Precisely,* * the table is constructed as follows. The elements (w00(1), . .,.w00(r0)) are the first terms* * of the permuta- tion (w0(1), . .,.w0(r)). The elements (w0i(1), . .,.w0i(ri)) are the first ter* *ms of the permu- tation (wi(1), . .,.wi(r)) which do not occur inside a line above in the table.* * Equivalently, we omit the values represented in the sequence w0(1), . .,.w00(r0 - 1), w01(1),* * . .,.w01(r1 - 1), . .,.w0i-1(1), . .,.w0i-1(ri-1- 1) from the permutation (wi(1), . .,.wi(r)). For instance, we have: fifi fi fi fi1, 2, 3, 4fifi1, 2,fifi1, 2, T R fifi1, 4,=3,f2ifi4,+ fifi4, 3, fi1, 2, 4, 3fi2, 4, 3fi2, 3. (We drop the degenerate terms, which are equivalent to 0, from the result.) We claim: 1.3.2. Theorem The table reduction morphism T R : E -! X defined in the Paragraph above * *is a surjective morphism of differential graded operads. The surjection operad X is * *a quotient of the operad E. The demonstration is achieved in the next Sections. Our purpose is to prov* *e both the Proposition 1.2.7 and the Theorem 1.3.2. In fact, as mentioned in the Theor* *em, we prove that X is a quotient operad of the classical operad E. The table reductio* *n morphism T R : E -! X is identified with the quotient arrow. Therefore, we prove the f* *ollowing Assertions first: 1.3.3. Assertion: The table reduction morphism T R : E - ! X is well defined. 1.3.4. Lemma: The table reduction morphism T R : E - ! X is surjective. Then, the next Assertion implies that the differential on X (r) verifies th* *e identity ffi2 = 0. Equivalently, we prove that X (r) is a quotient dg-module of E(r). 1.3.5. Assertion: The table reduction morphism T R : E - ! X maps the different* *ial in the Barratt-Eccles operad E(r) to the differential in the surjection operad X (* *r) (specified in Paragraph 1.2.3). The next Assertion is obvious from the definition of te table reduction mor* *phism: 18 1.3.6. Assertion: The table reduction morphism T R : E - ! X is equivariarant. The next Assertion implies that the composition product of the surjection o* *perad satisfies the properties of an operad composition product and is compatible wit* *h the dif- ferential. 1.3.7. Assertion: The table reduction morphism T R : E -! X maps the compositi* *on product in the Barratt-Eccles operad to the composition product in the surjecti* *on operad (specified in Paragraph 1.2.4). The announced conclusion follows: the surjection operad X is a quotient ope* *rad of the Barratt-Eccles operad E. The Assertions 1.3.3, 1.3.4 and 1.3.5 are proved in Section 1.4. The Secti* *on 1.5 is devoted to the demonstration of the last Assertion 1.3.7. 1.4. The operad morphism properties 1.4.1. Proof of the Assertion 1.3.3: If w 2 E(r)d is a degenerate simplex in the Barratt-Eccles operad, then the* * asso- ciated surjections w0 2 X (r)d are also degenerate. Therefore, the element T R* *(w) van- ishes in the normalized surjection operad. This proves that the table reduction* * morphism T R : E -! X is well defined. To be more explicit, suppose that w = sj(w0, . .* *,.wd) = (w0, . .,.wj, wj, . .,.wd). Consider a surjection w0 which arise from the table* * reduction of (w0, . .,.wj, wj, . .,.wd). In the associated table arrangement, the line j co* *nsists of the terms w0j(1), . .,.w0j(sj - 1), w0j(sj) of the permutation wj. The values w0j(1), . .,.w0j(sj- 1) are ommitted in the n* *ext row (by definition of the table reduction process). Therefore, this row (which consists* * also of terms of the permutation wj) starts with the term w0j(sj). We conlude that the last e* *lement of the line j coincides with the first element of the line j + 1. The Assertion fo* *llows. 1.4.2. Proof of the Lemma 1.3.4: Let u 2 X (r)d be a surjection. We specify a simplex w = (w0, w1, . .,.wd) * *2 E(r)d such that T R(w) = u. The permutations (w0, w1, . .,.wd) are subsequences of (u(1), * *. .,.u(r + d)) and are determined inductively from the table arrangement of the surjection. To be explicit, the table arrangement of u has the form: fifis(1), . . .s(i - 1),x(1), fifis(i ),. . .s(i1 - 1),x(2), fi . 1 .2 . u(1), . .,.u(r + d) = fifi.. .. .. fifis(i ),. .s.(i - 1),x(d), fis(i d-1 d d), . . .s(r), where (s(1), . .,.s(r)) is a permutation. We fix wd = (s(1), . .,.s(r)). To o* *btain the permutation wifrom wi+1, we just move the occurence of x = x(i) in the given su* *bsequence (wi+1(1), . .,.wi+1(r)) (u(1), . .,.u(r + d)) 19 to the position of the caesura x(i). Explicitly, we have: wd = s(1), . .,.s(r), wd-1 = s(1), . .,.s(id - 1), x(d), s(id), . .,.dx(d), . .,.s(r), wd-2 = s(1), . .,.s(id-1 - 1), x(d - 1), s(id-1), . .,.x(dd - 1), . * *. . . .,.s(id - 1), x(d), s(id), . .,.dx(d), . .,.* *s(r) and so on. The Lemma follows from the following Claim: 1.4.3. Claim: If (w0, . .,.wd) are defined as above, then we have T R(w0, . .,* *.wd) = (u(1), . .,.u(r + d)). The permutations wiand wi+1 coincide up to the position of x(i). To be more* * explicit: wi = wi(1), . .,.wi(p - 1), x(i), wi(p + 1), . .,.wi(q), wi(q + 1), .* * .,.wi(r), wi+1 = wi(1), . .,.wi(p - 1), wi(p + 1), . .,.wi(q), x(i), wi(q + 1), .* * .,.wi(r). Now, consider a term fifiw0(1), . . .w0(r - 1), w0(r ), fifiw00(1),. . .w00(0r - 1), w00(0r ), fifi.1 1. 1 1. 1 fifi.. .. .. fifiw0d-1(1),. .w.0d-1(rd-1 - 1),w0d-1(rd-1), w0d(1), . . .w0d(rd) in T R(w0, . .,.wd). The caesura in the permutation wi is necessarily put on x(* *i). Other- wise, in the table, there is a repetition between the lines i and i+1. Precisel* *y, one observes that the last term of the line i would coincide with the first term of the line* * i + 1. This property holds for any permutation wi. The Claim follows. 1.4.4. Proof of the Assertion 1.3.5: Fix w = (w0, . .,.wd) 2 E(r)d. Let w02 X (r)d be a surjection associated to* * w by the table reduction process. If we assume ri= 1, then we have: fifi 0 0 fifiw0(1), . .,.w0(r0), fifi... fifiw0i-1(1), . .,.w0i-1(ri-1), w0(1), . .,.w0(r + d) = fifiw0i(1), fifiw0i+1(1), . .,.w0i+1(ri+1), fifi.. fifi. w0d(1), . .,.w0d(rd). Clearly, in the differential ffi(w0) 2 X (r)d-1, the term, which drops w0i(1) f* *rom the sequence, is equivalent to a surjection associated to (w0, . .,.cwi, . .,.wd) 2 E(r)d-1 b* *y the table reduction process. Furthermore, in both cases, the sign in the differential is* * (-1)i. We 20 observe that the other terms in ffi(T R(w0, . .,.wd)) vanish.P Therefore, the d* *ifferential ffi(T R(w0, . .,.wd)) 2 X (r)d-1 reduces to T R(ffi(w)) = i T R(w0, . .,.cwi* *, . .,.wd). In fact, the ommission of an inner value in a surjection w02 X (r)d cancels* * the ommis- sion of a caesura in another surjection w002 X (r)d associated to w. Explicitly* *, assume that we drop the inner element w0i(p) from the surjection w0 2 X (r)d specified by t* *he indices r0, . .,.rd. As in the definition of the differential, we consider the previou* *s occurence of k = w0i(p), which is at a caesura w0j(rj) in the table arrangement of the surje* *ction. Then, let w002 X (r)d be the surjection determined by r0, . .,.rj + 1, . .,.ri - 1, .* * .,.rd. This surjection w00differs from w0 by the lines i and j of the associated table arra* *ngement: fifi 0 0 fifiw0(1),.. .,.w0(r0), fifi.. fifiw0j(1), . .,.w0j(rj - 1), k, w0j(rj + 1), w00= fififi... fifiw0(1), . .,.w0(p - 1), bk, w0(p + 1), . .,.w0(ri), fifi.i i i i fifi.. w0d(1), . .,.w0d(rd). We observe that the ommission of w0j(rj+1) in w00yields the same result as the * *ommission of w0i(p) in w0. Furthermore, the signs associated to these differentials are o* *pposite. (This assertion is an immediate consequence of the definition.) The conclusion follow* *s. 1.5. On the composition products The purpose of this section is to prove the Assertion 1.3.7 and to complete* * the demon- stration of the Proposition 1.2.7 and of the Theorem 1.3.2. Thus, we compare th* *e compo- sition product in the surjection operad to the composition product in the Barra* *tt-Eccles operad. 1.5.1. On the expansion of the composition products Fix elements u 2 E(r)d and v 2 E(s)d in the Barratt-Eccles operad. We prove* * that the composition products T R(u) Ok T R(v) and T R(u Ok v) agree term by term. E* *xplicitly, we have a map from the expansion of T R(u)OkT R (v) to the expansion of T R(uOk* *v). This map gives a one-to-one correspondence on the non-degenerate terms of the expans* *ions. The Assertion follows. Precisely, on one hand, in the expansion of T R(u Ok v), the terms are inde* *xed by the following elements: - a path (x*, y*), which fixes a composition (ux* Ok vy*) in the Barratt-Eccles* * operad; - indices (t0, . .,.td+e), which fixes a surjection (u Ok v)0 associated to the* * former compo- sition. By definition, these indices (t0, . .,.td+e) are characterized by the dimension* *s of the table arrangement of the surjection (u Ok v)0. On the other hand, in the expansion of T R(u) Ok T R(v), the terms are inde* *xed by: - indices (r0, . .,.rd), which fixes a surjection u0 associated to u; - indices (s0, . .,.se), which fixes a surjection v0 associated to v; 21 - indices 1 = j0 j1 . . .jn = s + e, which fixes a decomposition of v0. The associated composed surjection w0, which occurs in the expansion, is given * *by the insertion of the components of v0 in the fixed surjection u0. We analyze the table arrangement of the composed surjection w0 in the next * *Para- graphs. We observe that the insertion process determines a path (x*, y*) and a * *composite operation (ux* Ok vy*) in the Barratt-Eccles operad. The composed surjection w0* * is asso- ciated to the surjection (u Ok v)0 which results from the table reduction of th* *is element (ux* Ok vy*) and whose table arrangement has the same dimensions. 1.5.2. On the table arrangement of a composed surjection Thus, we need insights on the substitution process for surjections. We fix * *u02 X (r)d and v0 2 X (s)e as above. As in the Paragraphs 1.2.4 and 1.2.5, we let u0(i1),* * . .,.u0(in) denote the occurences of k in the sequence u0(1), . .,.u0(r + d). By conventio* *n, we have also i0 = 1 and in+1 = r + d. The surjection u0 is decomposed in (u0(i0), ____, u0(i1)) (u0(i1), ____, u0(i2)) . . .(u0(in), ____, u* *0(in+1)). Furthermore, we fix a decomposition (v0(j0), ____, v0(j1)) (v0(j1), ____, v0(j2)) . . .(v0(jn-1), ____,* * v0(jn)) of the surjection v0. These components are inserted in the decomposition of u0 (u0(i0), ____, u0(i1)) (v0(j0), ____, v0(j1)) (u0(i1), ____, u0(i2)) . . . . . .(v0(jn-1), ____, v0(jn)) (u0(in), ____* *, u0(in+1)) The composed surjection w0 is represented by the sequence obtained by concatena* *tion. Explicitly: u0(i0), ____, u0(i1 - 1), v0(j0), ____, v0(j1), u0(i1 + 1), . . . . .,.u0(in-1 - 1), v0(jn-1), ____, v0(jn), u0(in + 1),* * ____, u0(in+1). We omit to mark the substitution-shift for simplicity (precisely, we omit to in* *crease the elements v0(j) by k - 1 and to increase the elements u0(i) > k by s - 1). It is easy to determine the caesuras of the composed surjection: 1.5.3. Observation: The caesuras of the composed surjection above w0 are classi* *fied as follows. The caesuras u0x(rx) of the surjection u0 such that u0x(rx) 6= k give * *a caesura in the composed surjection. The caesuras u0x(rx) of the surjection u0such that u0x* *(rx) = k are the terms u0(i1), . .,.u0(in-1). The insertion process move such a caesura u0(i* *m ) to the last element v0(jm ) of the component (v0(jm-1 ), ____, v0(jm )) which is inserted o* *n u0(im ). The other elements (v0(jm-1 ), ____, v0(jm - 1), vd0(jm)), which are a caesura for * *v0, give another caesura in the composed surjection. Our next purpose is to relate the table arrangement of the composed surject* *ion to the composition procedure in the Barratt-Eccles operad. 22 1.5.4. The path determined by the insertion of a surjection We define the path (x*, y*) associated to a composed surjection. We let dm * * and em denote respectively the degree of (u0(im-1 ), ____, u0(im )) and (v0(jm-1 ), __* *__, v0(jm )). The definition implies that the sequence (u0(im-1 ), ____, u0(im )) intersects the * *lines d1 + . .+.dm-1 , d1 + . .+.dm-1 + 1, . .,.d1 + . .+.dm-1 + dm in the table arrangement of u0. Similarly, the sequence (v0(im-1 ), ____, v0(i* *m )) intersects the lines e1 + . .+.em-1 , e1 + . .+.em-1 + 1, . .,.e1 + . .+.em-1 + dm in the table arrangement of v0. By definition, the path (x*, y*) associated to the composed surjection is g* *iven by the concatenation of horizontal components of length dm , m = 1, . .,.n + 1, and of* * vertical components of length em , m = 1, . .,.n, as in the insertion process. The indic* *es (xc, yc) are given by the following formulas. Ifc = d1 + e1 + d2 + e2 + . .+.dm-1 + em-1 + x0, where 0 x0 dm , æ 0 then we have xcy= d1 + d2 + . .+.dm-1 + x , c = e1 + e2 + . .+.em-1 . Ifc = d1 + e1 + d2 + e2 + . .+.dm-1 + em-1 + dm + y0, where 0 y0 em , æ then we have xcy= d1 + d2 + . .+.dm-1 + dm ,0 c = e1 + e2 + . .+.em-1 + y . This definition is motivated by the table arrangement of the composed surjectio* *n. 1.5.5. Observation: In the component (u0(im-1 ), ____, u0(im )), the elements u0(i) 6= k, which* * come from the line x = d1 + . .+.dm-1 + x0 of the table arrangement of u0, are inserted i* *n the line c = d1 + e1 + . .+.dm-1 + em-1 + x0 of the table arrangement of the composed su* *rjection w0. In the component (v0(jm-1 ), ____, v0(jm )), the elements v0(j) which come * *from the line y = e1 + . .+.em-1 + y0 of the table arrangement of v0, are inserted on t* *he line c = d1 + e1 + . .+.dm-1 + em-1 + dm + y0 of the table arrangement of the comp* *osed surjection w0. (These assertions follow immediately from the classification of the caesura* *s in a com- posed surjection.) Now, we have: 1.5.6. Claim: The path associated to a composed surjection w0as above fixes a c* *omposition (ux*Ok vy*) 2 E(r + s - 1)* in the Barratt-Eccles operad. We claim that w0 agre* *es with the surjection (u Ok v)0 which results from the table reduction of (ux* Ok vy*) 2 E* *(r + s - 1)* and whose table arrangement has the same dimensions. As mentioned in Paragraph 1.5.1, we associate the composed surjection w0 in* * the expansion of T R(u) Ok T R(v) to the surjection (u Ok v)0 in the expansion of T* * R(u Ok v). 23 We prove the Claim above by induction. More precisely, we assume that the s* *urjections agree on the lines of the table arrangement such that xc < x, where 0 x d. * *Then: 1.5.7. Claim: The surjection (u Ok v)0 agrees with the composed surjection w0 o* *n the next lines of the table arrangement (on the lines such that xc = x). Proof: We assume that c0, c0+ 1, . .,.c00are the indices of the lines such that xc* * = x. We have: (xc0, yc0) = (x, y0), (xc0+1, yc0+1) = (x, y0+ 1), . .,.(xc00, yc00) = * *(x, y00). In the table arrangement of the composed surjection w0, the lines c0, c0+ 1* *, . .,.c00 either do not contain or contain an occurence of the surjection v0. In the fir* *st case, we have c0= c00and the line considered reads: |u0x(1), . .,.u0x(rx) In the second case, the lines c0, c0+ 1, . .,.c00of the table arrangement of th* *e composed surjection have the following form: fifi 0 0 0 0 0 0 fifiux(1),0. .,.ux(i0- 1), vy(j ), . .,.vy0(sy0), fifivy0+1(1), . .,.vy0+1(sy0+1), fifi ... fifiv000 (1), . .,.v000 (sy00-1), fiv0yy0-10(1), . .,.yv-10y00(sy00), u0x(i0+ 1), . .,.u0x(rx* *), where 1 i0 rx. It is also possible to have c0= c00. Then, the diagram above * *collapses to one line: |u0x(1), . .,.u0x(i0- 1), v0y(j0), . .,.v0y0(sy0), u0x(i0+ 1), . .,* *.u0x(rx), In the table arrangement of the surjection (u Ok v)0, the lines c0, c0+ 1, * *. .,.c00arise from the table reduction of the permutations ux Ok vy, where y = y0, y0+ 1, . .* *,.y00. The permutation ux Ok vy is represented by the sequence: ux(1), . .,.ux(p - 1), vy(1), . .,.vy(s), ux(p + 1), . .,.ux(r), where ux(p) = k. By definition, in this sequence, we have just to omit the ele* *ments, which already occur inside a line of the surjection. By the induction hypothes* *is, these elements occupy the same position in the table arrangement of the composed surj* *ection w0. Therefore, we conclude: the elements, which we have to drop from the permut* *ations uxOkvy, are the same as the elements which are ommitted in the next lines of th* *e composed surjection w0. This observation allow us to prove that the next lines of the s* *urjection (u Ok v)0 agree with the composed surjection w0. We follow the construction. 24 Let us assume that we have not to omit all elements vy0(1), . .,.vy0(s) in * *the construc- tion. As observed, the elements ux(q) which we have to drop from ux(1), . .,.u* *x(p - 1) are also ommited in the sequence u0x(1), . .,.u0x(rx). It follows that the fir* *st terms in- serted on the line c0 of the surjection (u Ok v)0 and of the composed surjectio* *n w0 agree. If the line is not completed, then the elements which we insert from vy0(1), . * *.,.vy0(s) and from ux(p + 1), . .,.ux(r) are same for the same reason. If c0< c00, then o* *n the lines c = c0+1, . .,.c00of the table arrangement of (uOkv)0, we omit ux(1), . .,.ux(p* *-1) because these elements occur either inside the line c0or still inside a line above in t* *he table. There- fore, the next elements that we have to insert come from vy(1), . .,.vy(s), y =* * y0+1, . .,.y00. These elements correspond again to the next elements v0y(1), . .,.v0y(sy0) of t* *he composed surjection w0. If we have to omit all elements vy0(1), . .,.vy0(s), then, for the table ar* *rangement of the composed surjection, we obtain: |u0x(1), . .,.u0x(rx). Furthermore, the elements u0x(1), . .,.u0x(rx) belong necessarily to the last c* *omponent (u0(in), . .,.u0(in+1)) of the surjection u0. Therefore, the value k has to be * *omitted from ux(1), . .,.ux(p - 1), k, ux(p + 1), . .,.ux(r) in the construction of u0x(1), * *. .,.u0x(rx). The omission of the block vy(1), . .,.vy(s) in the permutation ux Ok vy gives the s* *ame result. The conclusion follows in this case. The demonstration of the Claim is complete. The next Claim implies that the non-degenerate terms (u Ok v)0 in the expan* *sion of T R(u Ok v) are associated to a composed surjection: 1.5.8. Claim: Fix a surjection (u Ok v)0 in the expansion of T R(u Ok v). Fix x* * = 0, . .,.d. We assume xc = x, for c = c0, c0+ 1, . .,.c00. Consider the sequence formed by* * the lines c = c0, c0+ 1, . .,.c00of the surjection (u Ok v)0: fifi(u O v 0)0(1), . .,.(u O v 0)0(t 0), fifi(ux Ok vy0 )0(1), . .,.x(uk Oy v 0 c)0(t 0 ), fifi .x k y +1 x k y +1 c +1 fifi .. (ux Ok vy00)0(1), . .,.(ux Ok vy00)0(tc00). This sequence is equivalent to a concatenation: (u0x(1), . .,.u0x(i0- 1), k) (v0(j0), . .,.v0(j00)) (k, u0x(i0+ 1), .* * .,.u0x(rx)), otherwise the surjection (u Ok v)0 is degenerate. Proof: The property is immediate if c0= c00. We assume c0< c00. In the table, all * *lines except the last one have to end with a term v0y(j) and all lines except the first one * *have to start with a term v0y(1), otherwise the table contains a repetition. In fact, if the line y ends with u0x(i0), then all values ux(i), which come* * before u0x(i0) in the sequence ux(1), . .,.bk, . .,.ux(r), are ommitted in the next lines of t* *he surjection. If 25 k comes before u0x(i0), then all values vy(1), . .,.vy(s) already occur as a fi* *nal element in the table. Therefore, the elements vy(1), . .,.vy(s) are also ommitted in the n* *ext lines. As a conclusion, the line y + 1 has to start with the element u0x(i0) and a repeti* *tion occurs in the table. Now, if we assume that the line y -1 ends with v0y-1(j), then the elements * *ux(i) which come before k already occur as a final element in the table. Therefore, the lin* *e y starts with an element v0y(1). The Claim follows. 1.5.9. Claim: A composed surjection w0 associated to u0 and v0 is non-degenerat* *e if and only if the surjections u0 and v0 are non-degenerate. In this case, the compose* *d surjection w0uniquely determines the surjections u0and v0together with the decomposition o* *f v0which gives the composition. Proof: The first assertion is immediate from the insertion process for surjections* *. The degen- eracies u0(i) = u0(i + 1) such that u0(i) 6= k are not removed by the insertion* * process. So do the degeneracies v0(j) = v0(j +1). If the surjection u0has a degeneracy u0(i* *) = u0(i+1), such that u0(i) = k, then, according to our notation, we have i = im and i + 1* * = im+1 , for some m = 0, 1, . .,.n. In the composed surjection, this degeneracy is equiv* *alent to the repetition of the element v0(jm ). The sequence u0x(1), . .,.u0x(rx), where x = 0, . .,.d, is uniquely determi* *ned by the terms of the surjection w0which are on the lines such that xc = x. Explicitly, * *the sequence u0x(1), . .,.u0x(rx) is recovered by the following operations. The elements su* *ch that k w0c(i) k + t - 1 are replaced by k. The elements such that k + t w0c(i) are* * decreased by t - 1. The consecutive occurences of k are collapsed to one term and this ac* *hieve the process. (If the surjection u0 is non-degenate, then the repetition of k follo* *ws from the insertion process.) Similarly, the sequences u0y(1), . .,.u0y(sy), y = 0, . .,.* *e, are determined by the terms of the surjection w0 which are on the lines such that yc = y. In t* *his case, we withdraw the elements such that w0c(i) k - 1 or k + t w0c(i) and we decreas* *e by k - 1 the elements such that k w0c(i) k + t - 1. Consecutive occurences of a same* * value are also collapsed to one term. This achieve the demonstration of the Claim. The Claims above identify the non-degenerate terms in the expansion of T R(* *u) Ok T R(v) to the non-degenerate terms in the expansion of T R(u Ok v). To complet* *e the demonstration, it remains to compare the associated signs. By definition, in T * *R(u Ok v), the sign is determined by a shuffle of the horizontal and vertical components o* *f the path (x*, y*). In T R(u) Ok T R(v), the sign is produced by the insertions of the c* *omponents (v0(jm-1 ), . .,.v0(jm )) of the surjection v0 in the surjection u0. It is imm* *ediate from the definition that these shuffles are same. Therefore: 1.5.10. Observation: The sign produced by the insertion of a surjection in the * *composition process agrees with the sign determined by the path (x*, y*) associated to the * *composed surjection. The demonstration of the Assertion 1.3.7 is now complete. 26 1.6. The little cube filtrations on operads The Barratt-Eccles operad has a filtration F1W(r) F2W(r) . . .FnW(r) . . .W(r) by simplicial suboperads FnW(r) whose topological realization |FnW| is (as a to* *pological operad) homotopy equivalent to the classical operad of the little n-cubes FdD (* *cf. M. Boardmann, R Vogt [4] and P. May [20]). Precisely, the topological operads |FnW* *| and FnD are connected by operad morphisms |FnW| -~- . -~-!FnD which are isomorphisms in the homotopy category of spaces. The filtration on the Barratt-Eccles operad is introduced by J. Smith in th* *e article [27] in order to generalize the Milnor F K-construction and to provide a simplicial * *model for iterated loop spaces. The work of T. Kashiwabara (cf. [15]) proves that the sim* *plicial set FnW(r) has the same homology as the space of the little n-cubes FnD(r). The equ* *ivalence as a topological operad follows from the work of C. Berger (cf. [3]). The idea * *is to relate the filtration to a particular cellular structure on these operads. We have an induced filtration on the associated dg-operad F1E(r) F2E(r) . . .FnE(r) . . .E(r) such that FnE(r) is equivalent to the dg-operad formed by the chain complexes o* *n the spaces of the little n-cubes. J. McClure and J. Smith have introduced a similar* * filtration F1X (r) F2X (r) . . .FnX (r) . . .X (r) on the surjection operad (cf. [22]). Furthermore, J. McClure and J. Smith hav* *e proved that the dg-operad FnX is also equivalent to the little cube operad. In fact, i* *t is possible to adapt the methods of C. Berger to the context of the surjection operad. Thi* *s result implies that we have quasi-isomorphisms: FnE -~- . -~-!FnX . Our purpose is to prove the following Lemma: 1.6.1. Lemma The table reduction morphism T R : E - ! X preserves the filtrations. Furth* *ermore, the induced morphism Fn T R: FnE - ! FnX is a quasi-isomorphism of differential* * graded operads. We state a more precise result below. 27 1.6.2. The cellular structures We recall the definition of the cellular structure which occurs on the oper* *ads P = E and P = X . A cell is associated to a pair (~, oe) such that oe 2 r is a perm* *utation and ~ is a collection of non-negative integers Øx,y(~) 2 N (where {x, y} {1, * *. .,.r}). The permutation oe 2 r determines also a collection Øx,y(oe). Explicitly, the* * element Øx,y(oe) is the permutation of (x, y) formed by the occurences of x and y in th* *e sequence (oe(1), . .,.oe(r)) which represents the permutation oe. Equivalently, we assum* *e x < y. The permutation Øx,y(oe) is the identity permutation (x, y) if oe(x) < oe(y) and Øx* *,y(oe) is the transposition (y, x) if oe(x) > oe(y). The cellular structure is specified by sub- r-modules F(~,oe)P(r) P(r). T* *he filtra- tion is related to the cellular structure by the relation: X FnP(r) = F(~,oe)P(r). ~x,y 0, define an operad mo* *rphism ffl* \ - : E - ! *E. 3.2.10. Remark The operads X and Z are also equipped with such morphisms X -! *X and Z - ! *Z. Furthermore, we have a commutative diagram: E ___TR___//X__AW___//_ØZØ | Ø Ø | Ø Ø fflffl| fflffl fflffl *E ___*ff////__*X_*æ//_ *Z 42 The morphism X (r)* -! X (r)*-r+1 is very similar to fflr\ - : E(r)* -! E(r* *)*-r+1. Given u 2 X (r)d, we let fflr \ u 2 X (r)d-r+1 be the sequence such that fflr \ u = sgn(u(1), . .,.u(r)) . (u(r), u(r + 1), . .,.u(r + d)), if (u(1), . .,.u(r)) is a permutation of (1, . .,.r), and fflr\u = 0, otherwise* *. This cap product is an operad morphism ffl* \ - : X - ! *X and makes the left-hand square commu* *te as stated. The morphism Z(r)* -! Z(r)*-r+1 is due to V. Smirnov (cf. [25], [26]). * *Let oe : N*( n) -! N*-1( n-1) be the dg-morphism such that: æ oe( (i0, i1, . .,.in)) = 0(i1,- 1, . .,.in - 1),ifi0o=t0,herwis* *e. The tensor powers oe r : (N*( n) r)d -! (N*( n-1) r)d-r are equivalent to a morphism Z(r)d-n - ! Z(r)d-r+1-n . In fact, this morphism of dg-modules is a morphism of operads Z -! *Z. It is* * also straightforward to prove that it makes the right-hand square commute. 3.3. Some proofs In this section, we determine the structure of the chain complexes N~*(X) = N~*(S0), N~*( 1) and ~N*(S1) as coalgebras over the surjection operad X . The algebra structure of N~*(X), a* *s stated in the Theorem 3.2.4, follows from these calculations and from our definitions. To begin with, let us recall the following fact: 3.3.1. Fact: There are morphisms of pointed simplicial sets S0 - ! 1 - ! S1 wh* *ich induce the morphisms of coalgebras: N~*(S0) -! N~*( 1) -! N~*(S1). We have explicitly: N~*(S0) = F . (0), ~N*( 1) = F . (1) F . (0, 1) and ~N*(S1) = F* * . (0, 1). The morphism N~*(S0) -! ~N*( 1) takes (0) 2 N~0(S0) to (1) 2 N~0( 1). The morphism N~*( 1) - ! N~*(S1) cancels (1) 2 N~0( 1) and maps (0, 1) 2 N~1( 1) * *to (0, 1) 2 ~N1(S1). We make explicit the cooperations AW (u) : N~*(X) - ! N~*(X) r associated * *to a fixed surjection u 2 X (r)s-1. This surjection is represented by the sequence u = (u(1), . .,.u(s), . .,.u(s + r - 1)). The next assertion is immediate: 43 3.3.2. Fact: Fix a surjection u 2 X (r)0. In N~*(S0) r, we have AW (u)( (0)) =* * (0) r. As a consequence, in N~*( 1) r, we have AW (u)( (1)) = (1) r. Our next purpose is to determine the components of the coproduct X AW (u)( (0, 1)) = (C(1)) . . . (C(r)) in N~*( 1) r. In fact, the module N~*( 1) r is generated in degree s by the pe* *rmu- tations of the tensor (0, 1) s (1) r-s (for 0 s r). If u has degree s * *- 1, then AW (u)( (0, 1)) has degree s. The next Lemma gives the component (0, 1) s (* *1) r-s of AW (u)( (0, 1)), for 1 s r. This result suffices to determine all compo* *nents of the coproduct AW (u)( (0, 1)) by r-equivariance. 3.3.3. Lemma Fix a surjection u 2 X (r)s-1, where 1 s r. The coproduct AW (u)( (0,* * 1)) 2 N~*( 1) r has no component (0, 1) s (1) r-s unless the sequences (u(1), u(2), . .,.u(s)) and (u(s), . .,.u(s + r - 1)) are permutations of (1, . .,.s) and (1, . .,.r). In this case, we have: AW (u)( (0, 1)) = sgn(u(1), . .,.u(s)) . (0, 1) s (1) r-s in N~*( 1) r. As a consequence: 3.3.4. Fact: Fix a surjection u 2 X (r)r-1. The coproduct AW (u)( (0, 1)) vanishes in N* *~*(S1) r unless the sequences (u(1), . .,.u(r)) and (u(r), . .,.u(r + r - 1)) are both p* *ermutations of (1, . .,.r). In this case, we have AW (u)( (0, 1)) = sgn(u(1), . .,.u(r)) . (* *0, 1) r. The proof of the Lemma is postponed to the end of the section. We now deter* *mine the coproducts w* : ~N*(X) -! N~*(X) r associated to an operation w 2 E(r)d. Fi* *rst, let us record the following fact: 3.3.5. Observation: Fix 1 s r. If the surjection u 2 X (r)s-1 verifies the * *condition of the Lemma, then it has the following table arrangement: fifiu(1), fifiu(2), fifi. fifi.. fifiu(s - 1), u(s), . .,.u(s + r - 1). The next assertion follows from this observation. 44 3.3.6. Claim: Fix w = (w0, . .,.wd) 2 E(r)d. Assume d = s - 1. Let w02 X (r)d b* *e given by the sequence w0= (w0(1), w1(1), . .,.wd-1(1), wd(1), . .,.wd(r)). This surjection w0 arises by table reduction of w for r0 = r1 = . .=.rd-1 = 1 a* *nd rd = r. The coproduct AW (T R(w)) : N~*(X) - ! N~*(X) r, where X = 1 or X = S1, reduc* *es to the operations AW (w0) : ~N*(X) -! N~*(X) r. The Theorem 3.2.4 follows from this Claim, from the Lemma 3.3.3 and the Fac* *ts 3.3.2 and 3.3.4 above. It is straighforward to complete our calculations and to* * obtain the formulas stated. In fact, the tensor e(1) . . .e(s) c(s+1) . . .c(r)2 C(1) r, where 0 s r, is dual to (0, 1) s (1) r-s. To be more precise, because * *of the commutation rules, the duality pairing fi s r-sff e(1) . . .e(s) c(s+1) . . .c(r)fi (0, 1) (1) is equal to (-1)oe, where oe = s(s - 1)=2. 3.3.7. Proof of the Lemma 3.3.3: The surjection is represented by the sequence (u(1), . .,.u(s), . .,.u(s + * *r - 1)). We let (C(1)) . . . (C(r)) be the multi-simplex which is determined by the sequ* *ence 0 = n0 n1 . . . ns+r-1 ns+r = 1. To be explicit, we assume ni = 0, for i = 0, . .,.j - 1, and ni = 1, for i = j, . .,.s + r. By definition, the verte* *x (0) is a face of (C(i)) if i 2 { u(1), . .,.u(j) }. Similarly, the vertex (1) is a fa* *ce of (C(i)) if i 2 { u(j), . .,.u(s + r - 1) }. Furthermore, the simplex (C(i)) is degene* *rate if the index s occurs twice in u(1), . .,.u(j) or in u(j), . .,.u(s + r - 1). Therefor* *e, if the tensor (C(1)) . . . (C(r)) is equal to (0, 1) s (1) r-s in ~N*(C1) r, then (u(1), * *. .,.u(j)) is a permutation of (1, . .,.s) and (u(j), . .,.u(s + r - 1)) is a permutation * *of (1, . .,.r). This property may occur for j = s only and supposes that the surjection has the* * following table arrangement: fi fifiu(1), fifi... fifiu(s - 1), fiu(s), . .,.u(s + r - 1). Let us now determine the sign which occurs in the definition of AW (u)( (0* *, 1)). The intervals [ni-1, ni] (i = 1, . .,.s - 1), which are reduced to the point {0}, h* *ave length 1 and have 0 as a position sign-exponent. The other intervals are associated to a* * final-value of the surjection. Thus, the interval [ns-1, ns], which is equal to [0, 1], has* * length 1. The last intervals [ni-1, ni] (i = s + 1, . .,.s + r - 1), which are reduced to {1}* *, have length 0. As a consequence, the permutation sign associated to u is identified with th* *e signature sgn(u(1), . .,.u(s)). The formula of the Lemma follows. 45 References 1.M. Barratt, P. Eccles , On + -structures. I. A free group functor for st* *able homo- topy theory, Topology 13 (1974), 25-45. 2.D.J. Benson , Representations and cohomology II. Cohomology of groups and * *modules, Cambridge Studies in Advanced Mathematics 31, 3.C. Berger , Op'erades cellulaires et espaces de lacets it'er'es, Ann. Ins* *t. Fourier 46 (1996), 1125-1157. Cambridge University Press, 1991. 4.J. Boardman, R. Vogt , Homotopy invariant algebraic structures on topolog* *ical spa- ces, Lecture Notes in Mathematics 347, Springer-Verlag, 1973. 5.D. Curtis , Simplicial homotopy theory, Advances in Math. 6 (1971), 107-209. 6.A. Dold , Über die Steenrodschen Kohomologieoperationen, Ann. of Math. 73 * *(1961), 258-294. 7.P. Gabriel, M. Zisman , Calculus of fractions and homotopy theory, Ergebni* *sse der Mathematik und ihrer Grenzgebiete 35, Springer-Verlag, 1967. 8.E. Getzler , Cartan homotopy formulas and the Gauss-Manin connection in cy* *clic homology, in üQ antum deformations of algebras and their representations (R* *amat- Gan, 1991/1992; Rehovot, 1991/1992)", Israel Math. Conf. Proc. 7 (1993), 65* *-78. 9.E. Getzler, J. Jones , Operads, homotopy algebra and iterated integrals for* * double loop spaces, preprint (1994). Internet: http://arXiv.org/abs/hep-th/9403055. 10. V. Ginzburg, M. Kapranov , Koszul duality for operads, Duke Math. J. 76 * *(1994), 203-272. 11. P. Goerss, J. Jardine , Simplicial homotopy theory, Progress in Mathematics* * 174, Birkhäuser, 1999. 12. V. Hinich , Homological algebra of homotopy algebras, Comm. Algebra 25 (19* *97), 3291-3323. 13. V. Hinich, V. Schechtman , On homotopy limit of homotopy algebras in "K-t* *heory, arithmetic and geometry (Moscow, 1984-1986)", Lecture Notes in Mathematics * *1289, Springer-Verlag (1987), 240-264. 14. T. Kadeishvili , The structure of the A(1)-algebra, and the Hochschild and * *Harrison cohomologies, Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SS* *R 91 (1988), 19-27. 15. T. Kashiwabara , On the homotopy type of configuration complexes, in Ä lg* *ebraic topology (Oaxtepec, 1991)", Contemp. Math. 146 (1993), 159-170. 16. M. Livernet , Homotopie rationnelle des alg`ebres sur une op'erade, Thesis* *, Universit'e Louis Pasteur, Strasbourg, 1998. 17. S. Mac Lane , Homology, Die Grundlehren der mathematischen Wissenschaften * *114, Springer-Verlag, 1963. 18. M. Mandell , E1 algebras and p-adic homotopy theory, Topology 40 (2001),* * 43-94. 19. P. May , Simplicial objects in algebraic topology, Van Nostrand, New York, * *1967. 20. ________, The geometry of iterated loop spaces, Lectures Notes in Mathemati* *cs 271, Springer-Verlag, 1972. 21. J. McClure, J.H. Smith , A solution of Deligne's conjecture, preprint (199* *9). Inter- net: http://arXiv.org/abs/math.QA/9910126. 46 22. _________________________, Multivariable cochain operations and little n-cu* *bes, pre- print (2001). Internet: http://arXiv.org/abs/math.QA/0106024. 23. D. Quillen , Homotopical algebra, Lecture Notes in Mathematics 43, Springer* *-Verlag, 1967. 24. V. Smirnov , Homotopy theory of coalgebras, Izv. Akad. Nauk SSSR Ser. M* *at. 49 (1985), 1302-1321. English translation in Math. USSR-Izv. 27 (1986), 575-59* *2. 25. ____________, On the chain complex of an iterated loop space, Izv. Akad. Na* *uk SSSR Ser. Mat. 53 (1989), 1108-1119. English translation in Math. USSR-Izv. 35 (* *1990), 445-455. 26. ____________, The homology of iterated loop spaces, preprint (2000). Internet: http://arXiv.org/abs/math.AT/0010061. 27. J.H. Smith , Simplicial group models for n nX, Israel J. Math. 66 (1989), * *330-350. 28. J.R. Smith , Operads and algebraic homotopy, preprint (2000). Internet: http://arXiv.org/abs/math.AT/0004003. 29. N. Steenrod , products of cocycles and extensions of mappings, Ann. of Ma* *th. 48 (1947), 290-320. Mail address: Laboratoire J.A. Dieudonn'e, Universit'e de Nice, Parc Valrose, F-06108 Nice Cedex 02 (France). E-mail address: Clemens Berger , Benoit Fresse . 47