Higher products

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 148, April 1970 HIGHER PRODUCTS BY GERALD J. PORTERO The study of secondary product operations has played an important role in algebraic topology. Among examples of such operations are the toral construction of Toda [9] and the secondary product of Massey [4]. The first of these has always been defined topologically while the second has been defined algebraically. This difference has been carried over to higher order products where higher Toda brackets have been studied by Gershenson [1]; and higher Massey products, by Kraines [3]. Even in Spanier's general setting [7] Toda brackets are defined in a topological category and Massey products in an algebraic category. Massey products may, however, be defined in homotopy terms by means of a spectrum. In this paper we present a unified study of higher product operations in homotopy theory. This gives higher Massey products and higher Toda brackets as examples of the same general construction. It also extends the notion of Massey product to extraordinary cohomology theories with associative products. A similar construction is given for the commutator product in a loop space and higher commutator products are defined. It is shown that these are related to the generalized higher order Whitehead products defined in [5]. Given a set of based topological spaces, {F,};6/, we say an associative pairing, p: RiXRj-* RXJ(R(j=Rk some k eJ), is a product if p(x, *) = p(*,y) = *, where * is used ambiguously to denote the respective basepoints. Such a product induces an external product [X,Rx]x[Y, R,]->[Xx Y,Rtj] and an internal product [X, ÄJ x [X, Rj] -* [X, Rxj] where [X, Y] is the set of based homotopy classes of maps X —>Y. For homotopy classes ax e [Xl7 /?,], a2 e [X2, R¡], and a3 e [X3, Rk] a secondary external product {a1( a2, a3} e [/\ (A\, X2, X3), Q.Rm] is defined whenever the external products aia2 and a2a3 are zero. (/\ = smash product, Q = loop space). If Xx = X2 = X3 = X and the internal products a»^ and a2a3 are zero, a secondary internal product <a1; a2, a3} e [X, ClRiJk] is defined. Received by the editors August 2, 1968. O The author was supported in part by NSF GP-6969. Copyright © 1970, American Mathematical 315 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use Society 316 [April G. J. PORTER More generally an (n-l)ary K, external product ...,an}e[/\iXy,..., Xn), Qn~2Riiy,..., in)] and an (n —l)ary internal product <«!,..., O e [X, fi"" 2Riiy,...,in)] are defined whenever certain lower order products vanish. The external products are generalizations of the secondary products in stable homotopy and the internal products are generalizations of the higher Massey products. The construction is by means of a universal example and the same construction defines both products. We begin in §1 by giving some definitions and-notation. The higher products are then defined in §2. The properties of the higher products are studied in §3. It is shown that they are natural on the left with respect to any map (Theorem 3.1) and natural on the right with respect to morphisms of ringed sets (Theorem 3.2). If [£j, £¡] is a group, multiplication by an integer can be defined. It is shown that the products are linear with respect to this multiplication (Corollary 3.4). An associativity relation between higher products is stated and proven in Theorem 3.7. This relation enables us to view higher products as operations derived from higher associativity and to compute the ambiguity introduced in the last step of the construction (Theorem 3.9.). It is well known that cup products are zero in suspensions. Similarly the higher internal products vanish on suspensions (Theorem 3.10). From this it follows that the higher internal products are annihilated by loop suspension (Corollary 3.12). External products, on the other hand, are of great interest in suspensions. In Theorem 3.14 we show that the external products commute with suspension in an appropriate sense. Finally a relationship between the primary product and the higher product is given in Theorem 3.16. As a particular application of our construction we are able to define Massey products in any cohomology theory arising from an associative ringed spectrum. In §4 we define "cochains" for such a theory. Roughly speaking, cochains correspond to null-homotopies. If h is a null-homotopy of/, the coboundary S is defined by 8h =fi. Thus cocycles correspond to null-homotopies of the constant map, i.e. loops. Using these ideas we show that formally our definitions coincide with the definition of higher Massey products given in [3]. The construction given in §2 requires strict associativity of the product. If the pairing is only homotopy associative, the analogue of Stasheff's An forms [8] are required. We study this problem in §5 and indicate how to define n-fold products given a ringed set with An forms. Finally, in §6 we consider the commutator product in a loop space. In this case License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1970] HIGHER PRODUCTS 317 the products are nonassociative but do satisfy a Jacobi identity. A construction similar to the one made in §2 is given and higher commutator products are defined. Again there are internal and external products. The external product is a generalization of the Samelson product and is adjoint to the generalized Whitehead product defined in [5]. Four definitions are given of higher commutivity of a loop space, QX. If Zis an //-space these all agree; however, in general the precise relationship between these definitions is unknown. 1. Preliminaries. We assume throughout that all spaces have base points, denoted *, and all maps are continuous and base point preserving. The set of maps f: X-> Y will be denoted by {X, Y} and the set of homotopy classes of maps X^ y will be denoted [X, Y]. The cartesian product, Xx Y, is defined as usual. Xv Y is the subset of XxY with at least one coordinate at a base point. The smash product A iX, Y) = iXx Y)HXvY). The iterated smash /\ iXy,..., Xn) is defined inductively to be Ai/\(Xy,...,Xn_y),Xn). We note that any other grouping, e.g. /\ i/\ iXy,..., X,), /\ iX,+1,..., Xn)), is homeomorphic to f\ iXy,..., Xn). Definition 1.1. We say that M={R¡, p}je] is a ringed set of topological spaces if each R, is a topological space with base point and for certain distinguished pairs (/, j)eJxJ, p.: £¡ x R, —>Rk (some keJ) is a continuous map which sends £¡v£; to *. In general we shall write R(i,j) for the range of the multiplication on R¡xRj. Assume inductively that we have defined "distinguished /t-tuple" for k^n-l. We say an n-tuple, (£;i,.. .,Rin), is distinguished if the (n—l)-tuples, (£A,..., Rjn_y) and (£y2,..., £y„), are distinguished and the pairs, (£(/,.. .,jn-y), RjJ and iRh, £(/>,...,/))» are distinguished. If iRh,..., Rjn) is a distinguished n-tuple we insist that all possible iterated multiplications have R(Jy,.. .,jn) as their common range. Definition 1.2. âë is said to be associative if the following diagram commutes for all distinguished 3-tuples (/',/ k) and homotopy associative if the diagram homotopy commutes. £, x £, x Rk —-> £(/, j) x Rk Ixp RtxRiJ,k) M >Rii,j,k) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 318 [April G. J. PORTER Examples. 1. SÎ={SP, p}peZ+,ail pairs are distinguished, and p: SpxSq-> is the quotient map. This pairing yields the stable Toda product. 2. Sf=F({Xx}ieI), the free ringed set generated by {Xi}ieI, is {X(ilt..., = A C^ij.»■■■>^i„)} a^ Pa'rs are distinguished and p is the quotient map. Sp+q in) 3. S&={K(tt, m), p} where K(tt, m) is an Eilenberg-MacLane space, tt a commutative ring with 1. (K(tt, m), K(tt', n)) is distinguished if there is a pairing tt ® it' -*■tt" for some it". Then p: K(tt, m) x K(tt', n) -»■K(tt", m + n) is the cup product. In general p may not be strictly associative, however a model may be chosen for K(tt, n) and p such that p is associative. If p is only homotopy associative then for all n there exist An forms for p. (See §5 for definitions of An forms.) This pairing yields the classical Massey product. 4. M = {XY}where XY is the space of continuous base point preserving maps, F->- X, with the compact open topology. A pair (BA, Dc) is distinguished if C = B and, in this case, p:BA x DB-^ DA is composition. This pairing gives rise to the classical Toda bracket. 5. Given a ringed set SÎ define 0.3$ to be the set consisting of Q.'Rm for y 2:0 and Rm e Si. The pairing p.: ü.jRm x ClkRn-> Q.i+kR(m, n) is given by ß(Xx, X2)(tx, ■ ■ -, tj+k) = p(Xi(tx, ■ ■ -, tj), X2(tj+xi ■ • •; tj+k)) where xx e Cl'Rm, x2 e £lkRn and (for this definition only) we consider Cl'Rm as the set of base point preserving maps (V, I') -> (Rm, *) and p is the pairing given in S?. Clearly O.SI is a ringed set. (QM)k is the ringed set consisting of Q.iRmfor j^k and £ as above. 6. Given a ringed set Si define Sá? to be the set consisting of ^Rm fory'^0 and Rm e Si. The pairing p.: 2'Rm x SfcFn -*■Z' +kR(m, n) is the composite IfR, x XkRn-> A C&Rn,ZkRn)-*-2>+kA (*», Fn) -> ^ +fcÄ(m,n) where the last map is induced by the pairing in Si. Clearly T,Si is a ringed set. (ZSi)k is the ringed set consisting of S'Fm for j^k and p. as above. 7. For any ringed set Si there is an associated ringed set \J Si of matrices of Si. The elements of V Si are mn-fold wedges of spaces in SI. We denote VlSiSn R» ISiSm; by {Riiim.n-A pair, ({Rtj}m,n,{Äy}p>Q), is distinguished if (a) p=n, (b) (Fw, R'jk) is distinguished in ^ for all i, j, k, (c) p: R„x R'jk-> R¡k for all i, j, k. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 319 HIGHER PRODUCTS 1970] Define p.: {Rtj}m,nx {£!,}„,„ -►{R",)m,P,if the pair is distinguished, by ß(x, y) = Kx, y) e £"fc if (x, y) e Rlf x R'jk for some i, j, k, = * otherwise. It is easily seen that this is well defined and V is associative. The matrix Toda bracket in stable [11] arises from this pairing. 8. Let 3ft be a ringed set in which each of. £¡x£j-> £¡. The product p.: R¡ x R¡ ->£(/', respect to <j{if P-(°i(x, y), z)) = orlA>(x,z), p(y, z)) and 3i is an associative ringed set if p. homotopy defined by J. M. Cohen Ä,eJ is an //-space with sum j) is said to be distributive with /¿(x, o,iy, z)) = a^x, y), pLx, z))¡. The set 11^ of matrices is defined to be the set of mn-fold products of elements of 3ft. Denote nis¡sm:is;sn -R./by [£ti]m>n.A pair, ([/?,,]„,„ [R[}]P.q),is distinguished if it satisfies conditions (a), (b), and (c) of the preceding example. Let (<7¡)n:(£¡)n -> £t be some n-fold iteration of o¡. On distinguished pairs the product fi'- [£ijm,n x [PiAn.p "~*"[RiUm.p is defined to be the composite n *»*n R,*<-+ n *»**»-► n n^^ n #* i,;' k.l i.j.k i.k j i.fc where the first map is the product of projections, the second map is induced by the product pin ai and the third map is the product of n-fold summations. It is easily seen that fiix, *)=£(*, y) = *. If ip>,o) is distributive, o¡ is associative and abelian, and p is associative then fi is associative. If 3ftis the ringed set described in Example 3, the products arising from flSft are the matric Massey products studied by J. P. May [12]. 9. If G is a topological group or an associative //-space then the product p.: G x G —>G is not a product in our sense since pi*, 1) ~ 1. This operation should more properly be called a sum. However the commutator product is a product in our sense but it is not associative. (Higher commutator products are studied in §6.) Let In denote the cartesian product of the unit interval with itself n times. The boundary of/" is denoted by ln. £"<=/" and £nc/n are defined by Un = {ity,..., r») | r, = 1 for some /'}, Ln = {ity,..., tn) | t, = 0 for some i} Un is the set of upper faces and £" the set of lower faces. Obviously UnxLnxIn~1. The path space of X is denoted by PX and the loop space by fi X. We take as our model for £fin-1A' the set of maps, /: (/", £") -> (AT,*); fi-'A' is the set of maps /: (/n +1, £n +1)->(Ar, *). Each of these sets is given the compact open topology and the constant map is chosen as base point. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 320 G. J. PORTER Let/: A -^PL\n-1X. For 1 =i^n the ith upper face, i.e. f'(a)(tx,..., we define/': tn-i) = fi(a)(tx,..., Similarly if/: A -+ Q.nX,for l£i£n+i f'(a)(tx, ...,tn)= A -+PQn-2X U-i, 1, tit..., by restriction to tn-x). we define/': A-^PCP^Xby f(a)(tx,..., We say g1;.. .,gn+1 e{A,PQn~1X} this case we define [April U-u 1, tit..., rn). are compatible if (gj)i = (gi)l~1 for i<j. In G=Z(-iy 1= 1 +igie{A,n»X} by setting G(a)(tx, ■■-, h-x, 1. ti+1,..., tn+1) = gi(a)(tx, ■■-, ti-x, ti+1, ■■■, tn+x) and G(a)\Ln +1= *. The compatibility conditions ensure that G is well defined. This is addition in the sense of the homotopy addition theorem. If (Rm, Rn) is distinguished the product pairing induces the following pairings {X,FÜ'"1^} x{X, PQx^RA -+ {X,PCl'+*-iR(m, n)}, {X,Pil'-iRJ x{X, Rn}-> {X,PW-iR(m, »)}, {X, Rm}x{X, FQ'-1^} ->{X, Pa'-iR(m, n)}. This is given in the first case by setting ifg)ix)ih, ...,t1+k)= B2(m, n)(f(x)(tx, ..., tf), g(x)(tJ+1,..., t]+k)) and in the other cases by similar formulae. If (tx, . ■., tj+k) eL1+k then either (tx,..., t¡) eV or (tj+1,..., tj+k)eLk. In either case (fig)(x)(tx,..., ri+fc)= * and fig lies in the desired set. These products correspond to the definition of cup product on the cochain level. (For details see §4.) 2. Construction of the higher products—Associative case. The n-fold universal product is defined in a universal example. The construction of the universal example is conceptually easy; however, there are many definitions to be made and details to be checked(2). The universal example corresponding to the distinguished n-tuple (RJv ..., RJn), is denoted En(jx,.. .,./„). To define this we make use of intermediate spaces Et(jx,.. .,jn), 2 = i<n, which may be thought of as universal examples for /-fold products in n variables. In particular, E2(jx,..., jn) is Rhx--x Rin- The n-fold product is an (n—l)ary operation on n variables. In this notation two-fold products are the usual products in the ringed set Si and the product (2) We strongly urge the reader to follow the details by considering the case «=4. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 321 HIGHER PRODUCTS 1970] RjXRk^-Rij,k) is denoted by 02ij,k). £„(/,.. from the canonical path fibrations as follows : .,/„) is the fibre space induced En(jy,...,jn) IT "n - l(jl, ■• •, jn + i - 2) £»-lC/l. • • -,jn) —-► il ®n-n(ji,■■-Jn+i-a) £i-2Ul; • • •>£) 2 H Û"_a^C/t.-• ■Jn+«-a) 3 >nan_4Ä0i-"/"+i-3) : i=l n-2 1 riWfJl+l.Íl+í) ¿aO'l, • • •, jn) -=-> n-2 O ßÄ(/'i' £ +1' £ +2) 1= 1 n-1 n s2(ji,ji+i) i=i E2(ji,...,jn) Figure 1 >riÄo*'£+i) ni=i n-l where dkijt,.. .,ji+k_x) is the composite of a canonical projection, Ek(Jx,.. .,jn) -> Ekiji,.. .,jk+i-x) and the A>fold product in the universal example, £)cO/i, • • -,jk+l-l)- We now make this definition precise. Let 3ftbe a ringed set. For all distinguished m-tuples of spaces, iRh,..., Rin), m^2, we set E2ijx,.. .,jm) = Rhx ■■■ xRJm and 02ijx,j2): Rh x RJ2 -^ Rijx,j2) equal to the product in 3ft. Assume inductively that for 2^k<n and m£k we have defined Ekijx,..., jm) and ekijx, ...,jk): Ekijy,.. .,jk) -> W-2RiJx,.. .,jk) such that (a) if 1 ¿¡s^m, and k—l^t^m —s there exists a projection <ls,t'- £fcC/l> • • -,jm) ~~>Ekijs, ■• -,js+) such that the following diagram commutes for s^r<r + k—l^s Ek(jx,...,jn) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use + t. 322 G. J. PORTER (b) Ek(jx,..., jm) is the fibration nr--1fc+aQk-8ü01,...J,+fc-a)by m-fc-t-2 i= l induced [April from the path m-fc + 2 h-i(ji, ■■■Ji +k-2)gi.k-2- Ek-x(jx, ■. .,7m) -*■ Yl i= l (c) For l^i^k-l fibration over &k~3RiJi,■■;jt+k-a). and2^k<n (öfcO'l, • • .J*))* = Kjx,.. .Ji)-h(ji+x, ■■-Jk) where the product and face operations are as described in §1 and for t > 0 Kji, •••»;'(+¡): Ffcij'i»■..,jk)^P£ï-1R(ji, ...,ji+t) is the composite of fibre space projections FjcC/i, .. •, jk) -+ Ek_».(/i,...,/k)->-> Et+2(jx,..., jk) and the canonical projection Fi+2(/»,.. .,Á) -+PQ?-1R(Ji, . ■.,ji+t) (by condition (b)Fi+2c£i+1 x nFÜ'-^O",,...,;,+,))andh(ji): is the composite EAji, ...,jk)->-► E2(jx, ...,jk) EAj\,.. .,jk)-+Rit = Rhx-xRJk-+ h(ju ■■-Ji+t) is the canonical null-homotopy in EAJU ...,/,) product corresponding to Ru,..., R,,+t. Definition 2.1. For m^n define En(j\,.. from the canonical path fibration by *J-*: En_x(f\,.. .,jm) -> fi Rh. of the (r+l)-fold .,jm) to be the fibre space induced nn-3RiJi,. . .,ji+n-2) where •»m-1= m-n + 2 fi i=l dn-liJi,---,Jl +n-2)qf.n-2. The composite On_AJu .. .,jí+n-2)qlñ-2 is the map denoted 9n-%(jt, • ...Ji+n-a) in Figure 1 above. Thus En(jx,...,jm) xeEn-x(ji,---Jm) is the set of (m-n-|-3)-tuples and 0n-iC/i.■••Ji+.-Ä^W. t?, e FQ" " 3/?(/„ .. .,y',+n_2) (x,t¡x, .. .,r¡m_n+2) where is a null-homotopy to denote this.) For 0^s = m and n— 1 =t = m—s define by of (We shall write i?i(l)=<Vi(/.> .. .,ji+n-2)qlñ-2Íx) i?.«: FnC/i,.. .,jm)-> Fn(;s, ...,js+t) ^".¡(x»^i> ■• •>Im-n+a) = (??,rxW> -7s.• • •.■?»+«-»+a)- License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1970] 323 HIGHER PRODUCTS Using the inductive hypothesis on q^f1 one readily verifies that a£¡ satisfies the properties of (a) above. This completes the definition of £„(/,.. .,jm). Definition 2.2. The n-fold universal product corresponding 9nijx,.. .,jn): £„(/,...,/,) -* fi"-2£(/,. to £;i,..., Rjn, ..,jn) is defined by (0nfji,..., jn))'=Kjx,.. .,jt)-h(Ji+x,.. .,jn) for i=l,..., n-l and Uji, ---Jn) \Ln= *. The fact that this is well defined follows at once from condition (c). Clearly condition (c) is true for n. This completes the definition of the universal higher product. Definition 2.3. Given a map <p: X'-> Enijx,..., jn) define Mni<p), the n-fold Massey product (internal product), to be i<p)*[On]e [X, fin_2£(/,.. .,/„)]. We say <p:X-^EJjx,.. .,;'„) is of type 0/i, ...,/„) or equivalently <pis a lifting of X-► A /i x • • ■x/n Xn-1Î*. if n(/)<p~/, /= 1,..., n Rh x - - ■x Rjn where A is the diagonal and/: X-+ Rh. Definition 2.4. The set of n-fold Massey products of type ifi, ...,/„) </i, ...,/„> and definedby </i, • • •,/»> = {Mni<p) | <p:X-^EJJy,... is denoted ,jn) is of type ifi,... ,/„)}. For "nice" spaces z^iXyx • • • x Xn)~\J S /\ iXh,..., taken over all 1 ~ iy < i2 < ■■■< i¡ ^ n. Since Xtj) where the sum is [Xx x ■■■x Xn, fiT] = [S(Zi x • • • x Xn), Y] it follows that [f\ iXy,..., Xn), fiY] is a direct summand of [Xy x ■■■x Xn, fi Y]. Let a be the projection onto this summand. Definition 2.5. Given tp: Xy x ■■■x Xn -> Enijy,.. .,jn) the n-fold Toda product (external product) Tni<p)e [A (*i,..., Xn), CF-*Rifu .. .,jn)] is defined to be a(<p*[#„]). The definition of {fi,...,/,} which would be analogous to Definition 2.4 makes {/i. ■• -,fin} into too large a set, i.e. the indeterminacy of the construction is too great. Thus we consider only those maps <p: Xx x ■• • x Xn ->■£„(/,...,/,) which have the property that for each /c-tuple (/,..., i+k —l) with 2 = k<n and l = i ¿n—k+1 there is a map <p(z,..., i+k-l): XiX-x Xi+k_x -^ £fc+iO'i, • • -,jn) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 324 G. J. PORTER [April such that the following diagram commutes. 9 Xi X • • • X Xn ■ * Enijy, ...,jn) Kj\,---,ji+k-i) X¡ X ■• • X Xi + k_y <pii,...,i+k-l) Y Ek+yijy, . . .,/„)->PQ.k-2RiJi, . . .,ji + k-y) where the unlabeled arrows are canonical projections. We call such maps special. A map <p:Xy x ■■■x Xn -> EJJu ■ -Jn) is said to be of type (/1;.. .,fi) if <p is a lifting of fi x ■■■xfi: Xy x ■■■x Xn -» Rh x Definition x£, 2.6. The set of n-fold Toda products of type (/,.. .,/„) is denoted {fiu ■■-,fn} and is defined by ifu • • ■>/n} = {Tn(<p)| <p■ Xy x ■■■x Xn -> Enijy,... ,jn) is special of type (fi,... ,/„)}. All of the properties of {fi,.. .,/„} except Theorem 3.9 are true whether or not special maps are used. If the external product is defined so is the internal and we have : Proposition map. 2.7. à*{fi,.. .,/„}<=</,.. .,/„> where A: X-^- Xn is the diagonal The Massey product may, however, be defined without the Toda product being defined. Such, for example, is the case in singular cohomology. Finally we remark that for n>2, {fi,.. .,/„} and </1;.. .,/„> may be the empty set. 3. Properties of the higher products. In this section we study the properties of the products defined in §2. We note that A£,(<p)and £„(<?)are well defined elements of the appropriate homotopy groups while (fi,.. .,/„> andl/i,...,/,} are subsets of these groups. Thus for properties of Mn(<p)and £„(?>)in which we use " = " we must in general use "<=" for (fi, ...,/„> and {fi,.. .,/„}. Theorem 3.1 (Naturality). Letfi: Z-> Y and (a) let r- Y->EniJy,...,jr)thenf*Mni<p) = Mrii<pf)(b) Let yf.Y^ Rh, lúiún thenfi*(yy,..., ynyc(yifi ...,ynfi}. Let fi: Xi^- Yt, l=i^nand (c) let <p: Yyx ■■■x Yn-> £„(/,...,/,) then (A C/i, • ■-Jn))*Tn(<p)= Tni<pifiy x ■■■xfn)) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1970] HIGHER PRODUCTS (d) lety¡: F, -►Rh, l=i^n 325 then (A C/1,• • •,/n))*{ji, • • ■,yn}c {yifu ■■■,yJn} Proof. Clearly (b) follows from (a) and (d) follows from (c). To prove (a) we have f*Mni<p)=f*<p*ien]= i?fi)*l6„}= Mn(cpf). To prove (c) we note that the following diagram commutes. [T, x • • • x Yn, Q"-2Ä(A,. ..,;„)]-> (Fl x • • • x/„)* I A ( f [Xx x • • • x Xn, fl»-"*(/.,.. .JA] f Wt¡ [A (Yx,..., Yn),Q«-2RUu. ■.,/.)]-""" * > [A (Xx,..., XA,»-*MUu ■..,/.)] The proof of (c) is then immediate. Let Si and if be ringed sets. F: Si ->- if is said to be a morphism of ringed sets if F={f(j): Rj-^S, for je J} such that the following diagram commutes for all distinguished (i,j) eJxJ. -^SiXSj Rx x R, WU) m,i) fiU) * siUj) RiiJ) (3a) Theorem 3.2. Let F: Si -> y be a morphism of ringed sets and let 0%and 8%be the n-fold universal products corresponding to Rh,..., tively. Then there exists Fn : E%(jx,... ,jn) -> El (jx,..., such that esnFn= Q.-2(f(jx,.. .,jn))9*. Proof. We shall define gn¡k: E%(jx, ...,jn)-+ the following diagram commutes. Sntn E^(ji,...,jn) (3b) F£-i(ji, Rin and Sh,..., Sjn respecjn) which covers fh x ■■■xfin Eg(ju ■■.,jn),2 = kèn, ■+E*(jx,...,jn) .. .,j„)-—-► Rh X • ■• X Rin such that Ei_x(ji, ■■-,jn) fh x • • • x/>, lShx---xSin We then set gn,n = Fn. If n = 2 (3b) holds trivially and the theorem is simply the definition of a morphism of ringed sets. Assume inductively that gm>fcis defined form<n such that Fm=gm¡msatisfies the theorem. gn-2 is defined to be/yi x • • • xfijn. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 326 G. J. PORTER [April Assume inductively that gn.i is defined for 2^i<k and satisfies qi,tgn,i=gt+i,i<ll.t where 1 =s^n and i— 1 ^t^n —s and i<k. Denote the maps, constructions, etc. related to 3? by using super £ and those related to Sf by using super S. Recall that ( ESijl, n-fc+2 ■■■Jn) = \(x,r¡x,. y. ..,Vn-k-r2) e£?-l X | [ P^k'3R(ji, i=i ■■■,ji + k-2) such that ^¡(1) = 0j?_i(/„ .. .,yi+*-a)?i*fci-a(*)|. Define gn>fcby gn.fcO*, ^1. • • -,Vn-k+2) where ^=£fifc-3(/0'» ^(1) = (gn,k-l(x), fjl.'jn-k+is) • • ■Ji+*-a)X'7i)- = nk-3ifiji,.. .,ji + k-2))e^xiji,.. = "k-l(ji, ■ ■ ■, ]i + k-2)gk-l,k-lcii,k-2(x) = "k-l(ji, - - ■,ji+k-2)cIi,k-2Sn,k-l(x) .,ji+k-2)qtk-2(x) by the inductive hypothesis. Thus gn.k'- £fc(/'i, •••,£)-> £fc(ji. • -Jn)- This definition satisfies (3b) and the commutation formula with qk.t. We must still show that g„.n=Fn satisfies the theorem. Since £is a morphism of ringed sets, for each t, 1 ;£ t < n we have (fi-2/(/,. ..,jnm = PtP-XfUu.. .,jn)W(jx,. ..,jt)h*ijt+1,...,/)) = iPLY-2ifijy,...,jtWiJy,...,j)) ■iPW-'-2ifiijt+y, = . . .,jn)h»ijt+l, - - .,/)) (^gn.n)1- Thus the theorem is proven. Corollary 3.3. Let F: 3ft-» £f be a morphism of ringed sets (a) (QB"aA/l. • • -,Jn))*{Xl, ...,Xn}<= {fhXy, .. .,/,„*»} (b) (fi**- 2fijy, ..., jn)\(Xy, . . ., Xn) C (fihXy, . . .,/,„*„> where x¡ : Xt->- £¡ /n (a) and x¡ : A' -> £¡ /n (b). Remark. Homotopy commutativity of (3a) is not enough to ensure the existence of £„. We may define higher degrees of homotopy morphisms for £ which ensure that Theorem 3.2 holds. Alternately we say that £ is a homotopy morphism of ringed sets of order n if there exists £„ as above such that e*Fn~w-2fijx,...,jn)ei. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 327 HIGHER PRODUCTS 1970] If [/?,, F,] is a group for each Rxe Si we define k: Rt -> F, to be (Id-I-hid) (k times). We say Si is linear if the following diagram commutes for distinguished ft». /?, x R¡-> JR,x Rj (3c) Ri'J)-► k RiiJ) where the top map is either 1 x k or k x 1. Corollary 3.4 (Linearity). map kf: EAj\,...,jn) If Si is linear then for each t, l^t^n, there is a ->■E%(jx,.. .,jn) which covers 1 x • ■• x k x ■■■x 1 : Rh x ■■■x Rh x ■■■x Rin -> Rh x ■■■x Rh x • ■■x R,n and such that 6nkt=kdn. Proof. By enlarging the ringed set Si to include many (differently named) copies of each RxeSi we may assume without loss of generality that (/?,, R¡) is never distinguished. (It may of course be the case that for i^j, R¡=Rj and (R¡, R¡) is distinguished.) Define k¿j\, ...,jn): R(j\, ...,jn)-> kt(ji, ...Jn) R(j\,.. .,jn) by = k ifjs = jt for some s, = 1 otherwise. Under the above restriction it is easily seen that & is linear if and only if Kt = {ktijx, ■ -,/>)}: Si ^ Si is a morphism of ringed sets. The corollary then follows from Theorem 3.2 by noting that D.k is again k times the appropriate identity. Corollary 3.5. If Si is linear (a) let <p:Xx x ■■■x Xn -> EAj\,. ..,jn) be of type (fix,.. .,/„), then (kf)cp is of type (/i, ...,kfi,...,fn) and Tn(k^cp)= kTn(cp); (b){fix,...,kfi,...,fin}^k{fix,...,fin}; (c) let cp: X->En(jx,...,jn) be of type (fix, ...,/„) then (kficp is of type (fi, ...,kfj,... ,/„) and Mn(kfcp)= kMn(cp); (d) <fx,...,kfiJ,...,fny^k<fix,...,fin>. If Xi is an //'-space (e.g. suspension) then kfi is defined for/: Xx-> Ru and we may once again discuss linearity. Here the situation is easier and no hypothesis need be put on Si. Since Xx is an //'-space, A C^i> •••> X,,..., Xn) is also an //'space under /\(l,.. .,w,.. .,1) where w is the comultiplication in X¡. Thus A (1.k,..., 1) is k times the identity in A iXx, ■■-, Xn). The following is then a consequence of 3.1(c). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 328 G. J. PORTER [April Theorem 3.6. If cp: XxX ■■■x Xn^-EAJU .. .,jn) is of type (fix,.. .,/„) then cp(lx ■■■xkx ■■■x 1) is of type (fi, ...,kfi,.. .,/„) and (a) Fn(ç»(lx • • • x k x • • • x l))=kTn(cp), (b)k{fix,...,fn}^{fix,...,kfi,...,fin}. (We omit the corresponding results on Massey products since we show below that all Massey products vanish on suspensions.) Theorem 3.7 (Associativity). In [En.x(h, ■■-,/„), &n~3Rih, ■■.J»)l 0n-iiji, ■■.,h-i)-h(jA = (M)"+1n(j'iHn-i(J2, • • .,/„). Proof. In the notation of §2 consider nf(-l)t 1= 2 +1h(ji,---Jt)-h(jt+x,...Jn). This is a map from En_x(f\, ■■-,jn) to {(/»-1-lst and (n- l)st top faces), RKj\,.. .,;„)}. This can be considered as a map En-i(j1,...,L)->-(Q*~9R(jx,...,jjy is the desired homotopy. Corollary 3.8. Iftp: X-*E„-AJi, ■ -,Á) is of type (fi./„) which then 00 /i •Mn _ xigtn1-2<p)=i-l)n+% - ifeï.n1-2<P)/». (b) fix-if2, ...,/»> n (-l)n+1</!, ...,/„-!>•/„ fe nonempty. Let <p:Xxx ■■■x Xn -> i?»-iC/i,.. .,;„) èe o/rj/?e (A,.. .,/„) then (c) ine projections of /i-r„_1(ï3,-;1_29') [A (JTi,..., Jr„), Or-*R(ji,.. .,;„)] are equal. (d) /i-{/2, • • .,/„} n (-l)n +1{/i, • • -,/»-i}-/. a«^ (-l)n +1Fn-i(9Ï,n-29')-/n *'«ro Ö noneiiVO'. Proof. To see that (c) holds, one observes that the following diagram commutes. All maps are projections into direct summands. [Xxx ■■• x Xn,Sl"-3R(Ji,■-.,!•)]-► IA (Xx,..., X„-x)xX„, O'-'HUu .. .,/„)] [Xxx A (Jfa, • • •» x& n»-3R(j\,. ..,/„)]-> [A (Xx,..., Xn),a»-3R(ji,.. .,/„)] The proofs are then all evident. Hardie [2] has introduced the notion of a derived homotopy operation. Given a relation involving an (n-l)ary homotopy operation, one obtains a derived nary operation. In this language the secondary product is the operation derived from the associativity relation. Theorem 3.7 gives a higher associativity relation involving the (n —2)ary products. The (n— l)ary product 0n is then seen to be derived from higher associativity. This relation enables us to calculate the "modulus" of the construction at the last step. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1970] HIGHER PRODUCTS Theorem 3.9. (a) Let <?„_,: X-+En-l(ju...Jn) which can be lifted to En(Jx,.. be a map of type ifix,...,fi) ,,jn) then {Afn(<p)|<plifts <pn-i} & « coset of fx-[X, fi"-2£(y2,.. (b) Let 329 .,jn)]+[X, <pn-x: Xxx ••• xXn^En _i(/, (/i, • • -,/n) wA/cAcan be lifted to En(j\,. fi*1"2^/,...,Á-!)]•/„• ...,;'„) ..,jn) Ae a ioec/a/ wop o/ r^e then {£„(<p)|9>lifts <pn-y} is a coset of fi •[A (**,• • •>*»)>&n~2R(J2, - - -,jr)]+ [A (*i, • •., JT.-i).a—*Uu ■■•J»-i)]/n (vfAere íAese products are projected into [/\ iXy,..., Proof, (a) The lifting of <pn^xto £„(/,.. topies of Mn-yiq2~ni2(pn-y) Xn), Q"~2RiJx,.. .,ji)D- .,./'„) is given by choosing null-homo- and Mn-yiqy~n-2(pn-y). Any two null-homotopies of Mri-iiq2~n1-29n-i) differ by an element of [X, fi""2£(y2,...,/,)] and each element of this group can be expressed as the difference of two null-homotopies of A£,_1(a2,n1-2<Pn-i)-Similarly two null-homotopies of Mn_10?"pni2,r,n-i) differ by an element of [X, ün~2Rijy,.. .,jn-i)]- Thus if 9 and <p'are liftings of <pn-i (9>*[0»]-fy)*[«.]) efi■ [X, fi"-2£(;2,...,/)]+ [X, fi**-2£(/,.. .,jn-X)]-fi. (b) Because <pis special, the null-homotopies of Tt-MTn-m-ù and Tn-Xiqnx;nl2<pn-X) are determined up to elements of [X2x--x Xn, fi**-2£(;2,... Jn)] and [Xx x ■■■x Xn_x, Ün-2R(jx,... Jn-i)l respectively and the theorem follows as above. If q>were not special the homotopies in (b) would vary over [Xx x ■■■x Xn, fin-2£(;2,.. .Jn)] and [Xx x ■■■x Xn, Q.n~2R(jx,. ..,jn-x)] respectively and the cosets would be larger than above. This is the only property of the higher Toda products which uses "specialness". One notes that the more general problem of determining the "modulus" of (fi, ■■.,/»> or {fi,.. .,/„} is much more difficult. Theorem 3.10. Let <p:zZX'->• £„(/,...,/,) then Mn(r/>) = 0. Proof. We prove there exists y'~<p such that OJji, - ■-,jr)<p' = *- Let 9>r:SAr->£r(j'i, •..,/'„) be such that <prlifts <pr_xand <pn=<p.If r=2, <p2~(fi x ■■■x/„)An for some maps fit ■■-Jn, where An is the diagonal. Since I.X is a suspension A" can be factored through SXv • • • vEXfop to homotopy). Hence <p2is homotopic to the composite ZZ-»EJifv---vSZ f \ • -\f —-^Rhx---xRh. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 330 G. J. PORTER [April Call this cp22\Clearly 0aCÁ,,/i+i)?u9>22) = * for 1 ^i'^n— 1. By the homotopy lifting property there exist <pk2): ILX-* Ek(jx, ■■.,/'n)> 2^k^n, such that cpk2)lifts ç>J?2i and cpk2)~cpk. Since 82iJi,ji+x)<li,\<pT'' = *, <P32) is seen to be a map n- 1 2X^ E2(jx,...,jn)x H ßÄC/iJi+i)c ftCA,•••J») 1= 1 and hence is homotopic to a composite SI^SIV-VSI^ EAjx,...,jn)x H VRUiJui) i=l where the second map is <p(22)bi| ■• • hn-i with t¡x:I,X-> i2F(/,/+1). Call this map pi,3'. 03ÍJiJi-nJi+2)qi,2'P<3) is the "sum" of two products. Each of these products has as its factors two maps from 2 X which have the property that at each point of EX the image of at most one of them is different from *. Thus each product equals * and 63iJi,ji+x,Ji+2)<li,2<P33) = * for 1 ^i^n —2. By the homotopy lifting property there are maps <pk3): I,X-^- Ek(jx,...,/,) for k<3 such that ç43) covers tpjf-i and We continue in this manner until we define <pi,n) = <p'. By construction On(jl,-..JnW Corollary 3.11. Let fi: EA'-»- /?,, /'= 1,..., = *■ n, </l5.. .,/„> is /ne set consisting of the zero element. Corollary 3.12. Let Q,: [X, Q."-2R(jx,.. .,/„)] -* lax, Cl^RUx,.. be the loop map and let cp: X-^En(j\, .,jn)] ■. -,jn) then O*Mn(<p)= 0. Proof. Let A: ZÇIX-+ X be the adjoint of the identity on ¿IX. Then A*Mn(<p) = Mn(cpX)=0 by 3.1(a) and 3.11. Since Q* is the composite of A with the isomorphism [2QJST, &-2R(ji,.. .,/„)] ~ WX, W-iRVi, ■• -J»)] the corollary follows. Corollary 3.13. QEk(jx,.. .,jn)~flQ,t+1R(jx,.. taken over0 = t = k—2 and 1 = i^n —t. .,ji+t) where the product is We note that since the projection [Xxx---x Xn, Q»-aÄC/i,...,/,)] -> [A (*i, • • -, *0. U»-3ÄC/ls.. .,yn)] License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1970] HIGHER PRODUCTS 331 is not induced by a map of topological spaces there does not exist a functorial map to make the following diagram commute. [XyX---X Xn, fi"-2£(7i, . ...A)] -U [ÜiXyX---xXn), fi""1^/, . . .Jn)] 1 [A 0*1,. • -, Xn),n»-»R(Ji,. . .Jr)]-Î-* [ß A(Xy,..., Xn),fi""1^/, . . .Jn)] Thus although Q.*6n=0 it is not true, in general, that Q.*Tni<p) = 0 for tpl Xx X • • • x Xn -> £„(£, • • . Jn). The essential point in the proof of 3.10 was that cp2was defined using the diagonal in SA' and this can be factored through the one point union. Since this is not true for £„, it is not true that Tn vanishes in suspensions. On the contrary £n is of great interest on spheres where it represents the stable Toda bracket. We now study the stability properties of higher products. In what follows we consider the suspension homomorphism, 2»:»1I(íyjr)-».ir.+1(£MBjr), to be induced by Z(/)(M)(A) = (r,/0s)(A)) where t el, se Sn, XeI1. This is the composite nni&X) X TTn+ iiX)^nn+j+y(i:X) « nn + 1iVXX). Let 3ft be a ringed set and ~î*3ft the associated ringed set defined in Example 6 of §1. Let Erijy,...Jr), K(ji,---Jn) and Er(Zh, ...,SJb), universal examples and maps associated with Rh,...,Rin respectively. 0n{I.h,..., Z,B) be the and ~LRh,... ,~LRU Theorem 3.14. Let <p:Xyx ■■■x Xn -» En(Ji, ■■•,£,) « map of type 0/i,... ,/„). There exists a map oin, n):{XyX---x Xn, £„(/, ...,/,)}-> such that oin, n)i<p) is of type (E/i,..., Proof. For each r, k, and i,2^r^k rir, k, i): {Xyx---x Xk, £i'-aü(/,,.. {ZXy x ■■■x ZXn, £„(S,i;..., S,n)} Ifin) and Tnioin, n)itp))='LlTni<p). and l^i^k—r—l define .,/,+r_i)} ^{XXy x • ■• xZXk, fir-2S'£(;„ ..., Ji+r_i)} by (r(r, k, i)it))ity, Xy, t2, X2, . . ., tk, Xk)iX) = (f(, . . ., tl + r-y, ifliXy, ..., License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use xk)iX)) 332 G. J. PORTER where txel, x, e X and Ae/r_2. diagram commutes. [April A direct calculation shows that the following {Xxx • • • x Xk, Q'-'RUu ■■.Ji+,-i))-> ■Ar, k, i) {A (Xh . ..,Xt+r-x), &-2RUi, ■■•Jl+r-l»—HA (ZXx x • ■• xZXk, Q'-aI.'R<J{,...,/,+,-».)} (SJT„. . ..ZXi+r-A tl'-^'RUi.jt*,-l)) where the unlabeled arrows are projections. Thus to prove the last assertion of the theorem it suffices to show that for integers r and k, 2^r^k, there is a map o(r, k) such that the following diagram commutes for each i, 1 = i'=k—r —1. [Xxx.-.x X„, EAJu ■■ JA)-—-► {SJTix • • • xS*k, £,(Sil;.. .,£,*)} 0r(Ei(, . . ,,E/(+,_j. )#..-! 9,(Jl,---Jt*r-l)g,t.r-l {*, x • • • x A-,, n-^O), If A:=2 or r=2 r(r, A:,/) ■■.,/i+r-i)}-► (S* x ■■. xSATk,Ü'-2S'.R0,, • ■.,/i+r-j.)} the existence of a(r, k) and the commutativity diagram follow from the definition of ZSi. of the above We assume inductively that o(r, k) is defined so that the above diagram commutes if k<n and if k=n and r<m. Given <f>: Xxx ■■■x Xn-* EJJi, ■■•,/») let if>0:XxX---xXn-+ Em_x(ji, ■■-,jn), «/.,: Xx x ■■■ x Xn->PD.m-3R(ji,. for 1 -i^n—m+2 f(m-l, ..,ji+m-x) be the compositions of \\sand the canonical projections. Let n, i): {Xx x ■• • x Xn, />Q»-3lttj„ be defined similarly to t. -HS*, .. .,ji+m-i)} x ■• • xsin,pa-as-iRü,.. .,;,+m_»)} For any spaces C/and F let e: {£/, FF} -> {[/, F} be evaluation at the "endpoint". Then r(m-l, n, i)(e(cbx)) = e(f(m-l, n, /)(</<,)). Define a(m, n)(if>)by (a(m, n)(i/t))0= o(m —l, n)(\¡¡) and (o(m, n)(</i))i= f(m-l,n,i)(ipi), 1 ^ / = n—m+2. e((a(m,nM)d = e(f(m-l, n, i)(4>d) = r(m-\,n,i)(e(4>x)) = r(m-l, n, i)(&m-xijh ■■■,ji+m-2)qr.m-2>t') = em_xVh,..., £yi+m+2)?,m-V(AH-1, ")(</<) = 0m-xVlt, ■...^m-Mm-Mm, and thus o(m, n)(<p): ~LXxx • • • xSI„ -> Fm(Syi,..., n)>b)0 Sin). It remains to show that License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1970] HIGHER PRODUCTS 333 the diagram commutes. To see this it suffices to consider the case m=n. Thus we must show that the following diagram commutes. {XyX-.-x Xn, Eniji, - ■-Jn)}-a(n,n) > {ZXyX-.-x SZn, En(Zh,..., 0,(2*,..., 2,.) 6n(jl,- --Jn) {Xyx LJ} x Xn, fi"-2£0"i, • • •, fi)} T "' "' l> {SXyx - ■- x 2Xn, fi"-2S**£(/,..., /,)} We check this for the "pth face" of 0n. Let wt: 2?t->•£',_i be the projection. Let A^,..., Sebeas in §2. A(S;i, . . ., 2,p)(er(n, n)i>/>)) = Hp+l, = f(P,n, n)inp + 2TTp + 3- ■-TT^y l)((irP + 2---'r»0)i) = f(P,n, i)(h(Ji,---JP)>l>)Similarly n(Zjp +x, ...,SJ(a(n,n)(</-)) = fin-p,n,p+l)hijp+1,.. .,jn)i/j. A direct calculation using the definition of zZ3$shows (f(p, n, 0(A(/'i,. • -JP)4>))-(TÍn-p,n,p+l)ihijp+1,.. .,./„)</<)) = fin, n, l)ihijy,.. .Jp)-hijp+y,.. .,jn)4>). Since this holds for each p we have 0n(2yi,..., 2,>(n, "X*)) = <«. «>l)(Ön(/i, • • -Jr)<p)- This completes the proof of the theorem. Corollary 3.15. In z^Sftif{fy, ...,/„} is defined then Vl{fi,...,fn}^Wl,-..,mFinally we study products involving higher products. Theorem 3.16. (a) Let <p:X -+ £„(/,.. g: X-+ Rt. There exist maps 9V X-+Eni]i,..., ijn, 0) .,jn) be a map of type ifi,... and S<P-X-+Eniii,jy),j2,.. of type (fi,...,f„-g) and ig-fi, ...,fin) and Mnigcp)=g-Mni<p). (b) Let <p:Xyx ■■■x Xn -*■£„(/,.. respectively. .,;„) Moreover Mni<pg)= Mni<p)g .,;'„) be a map of type ifi,.. .,/„) and let g: T-> £¡. There exist maps <pg:XyX ■■■x/\ iXn, Y) -+ Enijy,..., gy: ¡\(Y,Xy)x---xXn-+ ,/„) and let (Jn, i)) and £„((/,/),.. .Jn) of typeifi,..., fi-g)and\g-fi,.. .,/„) respectively. Moreover Tni<pg)= Tnicp)-g and Tnig<p)=g-Tni<p). iWe consider the product to be from the smash product in (b).) Proof. The proof consists of multiplying the given liftings on the left or on the License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 334 [April G. J. PORTER right by g when they involve the first or last coordinate. The proof proceeds inductively and is straightforward. We omit the details. Corollary (a) </»,.. 3.17. Letfi: X-> R,j= 1,..., n and g: X-+ Rx then .,/„)'gc</i, ■..,/»•£>, (b)g-</i,...,A>c<g-/i,...,/„>. Letfi: Xj -> Rj,j= 1,..., n andg: Y^~ F, then (c) {fix,■■•,/n}-gc{/i, • • -,fn-g}, (d)g-{fx,...,fi}<={gfx,...,fn}. 4. Chains for cohomology theories. Often in mathematics the motivation behind a definition is obscured by the time the definition and the ensuing theory appear in print. In this section we present the background which led us to formulate the definitions given above. In so doing we give the relationship between our work and that of Kraines. In [3] Kraines defined higher Massey products in ordinary (singular) cohomology Our original goal was to extend these definitions to other cohomology theories. On "nice" categories cohomology theories are representable, i.e. given a cohomology theory Afn(X) there exists a spectrum, {YA, such that J^n(X) = [X, Yn] for all n. Thus cohomology classes correspond to homotopy classes of maps X-> Yn and the set of maps {X, Yn}can be thought of as the "cocycles" of J^n(X). Now, two cocycles are cohomologous if their difference is a coboundary. Similarly two maps are homotopic if their difference is null-homotopic. Following this idea we define the n-cochains to be the set of null-homotopies, {X,PYn+1}. The coboundary is evaluation at the endpoint. Thus the set of cocycles is {X, i)Fn+1} which we assume equal to {X, Yn}. Under the correct definition of "cycles mod boundaries" we then have 3fn(X) = [X, YJ. Furthermore if {YA is a ringed spectrum, this operation can be used to define the cup product of cochains. In this setting our definition of the higher Massey product formally coincides with the definition given by Kraines. For the sake of convenience we assume that the spectrum, <&= {YA has the property that ClYn+1= Yn for all n. Definition 4.1. Cn(X, <&)={X,PYn+1}. 8: Cn(X, W) -» Cn+1(X, <&)is induced by the composite: FFn+1-A- Yn+1 = QYn+2<=PYn+2 where e is evaluation at the endpoint. Clearly 82= * (the constant map). Set %n(X, ^) = ker 8 and SSn(X,$0 = lm S. %n(X, <&)is therefore equal to {X, Q.Yn+1}= {X, Yn}.Using the //-space structure of ÜFn+1, 2£n(X, <%)is a monoid with an operation corresponding to the additive inverse. Definition 4.2. z, z' e ¿£n(X, <W)are said to be equivalent, z~z', if there exists a sequence of cocycles z = z0, zl5..., zn = z' such that i=l,...,n. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use zx—zx-x&^n(X,'¥) for 335 HIGHER PRODUCTS 1970] Definition 4.3. Hn(X, <3/)= 3Tn(X,tW)j~. One easily sees Proposition 4.4. Hn(X,<W)= lX, Yn]. When constructing homotopy operations it is often useful to consider Cn(X; &) ={X,PQ.kYn+k+1} for k>0. To define the coboundary choose a relative homeomorphism, w: (In~\ F1"1) -» (/", Ln). This induces w: {X, ß""1 Y)-*{X, PQn~2Y} given by (wfi)(tx, ...,tn-x) =f(x)w(tx,..., r„-i). S' is then the composite e {X, FQ"-1 Yk}—> {X, ii""17J where e is induced by restriction. For fe{X,PQ.n~1Yk}, (f\...,fin) S7=wGí(-i)i+yO- ^ w {Z, FQ»~2 7,} is a compatible set in {X, PQ.n~2Yk} and Now assume $^={ Yk}is an associative ringed spectrum. As in §1 this pairing can be extended to give a product on the "cochain" level. Since (f-g)i=fi-g for i^n and (fi-g)'=f-g1~n for j>n we have on a formal basis ZWHf-sn = M;((|(-1)i+1/i)^)+M'(T(-1)i+y-(g)i-n) = (S'/)g+(-l)B/(8'g). A lifting <pof X ^>- Rhx ■■■x Rjn of type (ult..., wn) to En(jx,..., jn) is a choice of null-homotopies in X, i.e. cochains. Let h(jx,.. .,jk) be as in §2. The set of nullhomotopies {«(/, • • -,Jk)<P\l Û i Ú k = nand (i, k) ¥=(1, n)} corresponds formally to what Kraines calls "a defining system for the (cochain) product (iix,..., m„>" (Definition 1, p. 431of [3]). Under this correspondence our definition Mn(cp) = S(- l)k + \h(jx,. . .,jk)<p)-(h(jk+x, ■..,h)?) is formally the same as the definition given by Kraines. 5. Construction of the higher products—Homotopy associative case. The construction of the universal higher product in §2 depended upon the fact that the pairing of the ringed set was strictly associative. If the pairing is only homotopy associative a different construction is necessary and, in fact, one may be unable to to define higher products. Whereas in the associative case the construction consisted of 'gluing together' (n-1) maps, the construction given in this section License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 336 G. J. PORTER [April consists of'gluing together' 2"_1-1 maps. We present the outline of the general procedure. Stasheff, in his study of homotopy associativity of //-spaces [8], introduced the notion of An forms and An spaces. These forms measure the degree of homotopy associativity of an H space, G, and are the obstructions to constructing a classifying space for G. The notion of An forms carry over to ringed sets mutatis mutandis. We recall some details. The spaces Kn, n = 2 on which the An forms are defined are constructed as follows. Consider a word with / letters, xy ■■x¡. Corresponding to each nontrivial insertion of parentheses (in this word) there is a cell on the boundary of £¡. If the parentheses enclose xk- ■-xk+s.y we regard the cell as a homeomorph of KrxKs ir+s=i+1) under a map which is called 3fc(r,s). Two such cells intersect only on their boundary and the cells of the intersection correspond to the insertion of a second pair of parentheses. Start with £2 = *. Given £2,..., Kt_y construct the boundary of £¡ by fitting together copies of Kr x Ks as indicated above. Set £¡ equal to the cone on the boundary. Proposition Definition 5.1 (Stasheff). £¡;x£~2. 5.2. A ringed set 3ftadmits an An structure if there are maps Mi : £¡ x £;i x ■• • x Rh -> £(/,...,/) for 2Sifkn and all distinguished /-tuples (j\,.. .J) such that (1) M2 = p.: * x £¡ x R¡ -> £(/, j). (2) For PeKr,oeKs Miidkir, s)iP, a), Xy,..., (r+s = i+1) x) = Mrip, Xy, . . ., Xk-y, A£,(<7, Xk, . . ., Xk + S_y), Xk+S, . . ., X¡). Such a system of maps is called an An form on 3ft. A ringed set together with an An form is called an An ringed set. An A3 ringed set is a ringed set 3$ = {£,, p.} together with a homotopy M3: p-ipxl) ~ pilxfi). If 3ft is an An ringed set, n-fold products may be defined. The universal examples £r(/'i, • • •, jn) are defined as in §2. The change here is in the definition of 6n(Ji, ■■-Jn)- We first construct model spaces W¡ such that the set of maps/: Wt -> R under the compact open topology is our model for fi,_2£. Let Xi • ■•x„ be a word with n letters. Corresponding to each nontrivial partition of Xy■• ■x„ there is a cell on the boundary of Wn. In particular if Xy■• ■x„ is partitioned into t parts with r¡ letters in the z'th part, the corresponding cell is a homeomorph of Kt x Wri x ■■■x Wrt under a map S(r1;..., r). The cells on the boundary of £¡ x WTlx - - - x Wrt correspond to two operations on the partition of Xi • • ■x„. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1970] HIGHER PRODUCTS 337 (a) a nontrivial insertion of parentheses subject to the condition that the parentheses may not further subdivide any part, and (b) further subdivision of a given part with the subdivided part contained in parentheses. Example. x1|x2x3x4|x5|x6x7. Cells of type (a). (Xi IX2X3X4)IX51XgX7 XyI(X2XßX41X5)IXßX7 XyIX2X3XtI(X5 IX6X7) (Xi IX2X3Xi IX5) IXgX7 Xi I(X2X3X41X51X5X7) Cells of type (b). ^11(X% IXgXi)IXiIXgXi XyI(X2X31X4) IX51XgX7 XyI(X2[X31X4)IX51X6X7 Xi | X2X3X41X51(Xö | X7). The cell of type (a) correspond to the cells contained in Kt x Wri x ■■■x Wn while the cells of type (b) are contained in \J Kt x Wn x ■■■x WTlx - - ■x W,t. Two cells in the boundary of Wn meet only on their boundaries. Here two cells are identified if their related words (with partitions and parentheses) agree. Explicitly, we identify Kri, ■■-, rm+n)(di(m, n)(o, P), Xy,..., xm+n) with a(rx,..., r¡-y, R, ri+n,..., rm+n)(a, Xy,..., x¡_!, v(rit • • •) ri + n+y)ip, where £=2't"_1 X(, . . ., Xi + n-y), Xi + n, . . ., Xm + n) rt. Start with Wy= *. Given Wy,..., Wn-y construct the boundary of Wn by fitting together homeomorphs of Kt x Wri x - - - x WTtas indicated above. Set Wn equal to the cone of the boundary. We assert that Wnizln'x. We have checked this for n^5. To verify this for all n one must construct a model for Wn. Since there are 2n_1 —1 cells in the boundary, each of which is itself complicated this is a nontrivial combinatorial problem. We indicate below the construction for n ^ 4. (See figure on following page.) Define9JJU...,/): £n(/,... J„)~> (R(h, ■--,jn)f» by «U/i, • • -Jn)(x) diry,..., where (A:,oy,..., r)ik, oy,..., o) = Mlk, hijy,. ..Jry)(x)(oy),..., A0/n_r(+1,.. .,jn)(x)(a)) o) e Kt x WTlx - ■■x Wu and h(Jt,... ,js) is as in §2 and we understand A(/)(x)(*)=A(/)(x). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 338 G. J. PORTER [April Wx Xx\X2 W2 Xx\X2\X3 Xx\X2X3 XxX2\Xn w3 Xx\x2x^Xi XxX21x3x± Xx\X2X3\X4 Xx | X21X31X4 XxX2X3\X$ WA*) The verification that ön is well defined is straightforward and follows from the definition of the An forms and the construction of the Wxs. Mni<p), Tn(cp), (.fix,. ..,/„>, {fix,. ..,fi} are defined exactly as in §2. Theorems 3.1, 3.6, 3.7, 3.9, and 3.10 carry over to this case with no difficulty. The definitions of morphism of ringed sets and linearity must be changed to appropriate definitions in the category of An ringed sets. Under these new definitions Theorems 3.2, 3.4, and 3.14 hold. We leave the details to the reader. 6. Higher commutator products. Since the commutator product <1, 1>: QXxQX->ClX is neither associative nor homotopy associative, the constructions of §§2 and 3 are (3) All hidden faces are identified to * the base point. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1970] HIGHER PRODUCTS 339 not applicable. In this section we construct higher commutator products in loop spaces. The methods used here are a continuation of those used in the earlier parts of this paper. A different approach is taken by Williams [10]. Set Q.nY={f:In +1 -> Y), ÜY={fi: (I, Ï)->(Y, *)} and PCTY={fi:In+1 -> Y). We recall that the commutator, <1, 1>: ÙXx ÇIX-* QZis defined by <l,l>(/i,/2)01,?2) = /101) if '2 = 0, 1, = l2(t2) if tx = 0,l. Throughout this section we shall use many copies of QfX. To distinguish between them we shall index them, e.g. Q'X(kx, ■■-, kf). It will be clear from the context which copy is referred to. 02(i,j): QX(i)x ClX(j)-+ QX(i,j) is the commutator, <1, 1>, defined above. Set E2(X) = ÜX(l) x • • • x ÜX(n) and let q(i,j): E2(X) -> ÛX(i) x QX(j) he the projection. We assume inductively that Ei(X),jfik is defined for k<n and for j<m when k=n, and d,: E',(X)-+ £2i-1JSrisdefined for j<n such that: (a) For each /tuple (ilf ...,if), lái'i< ■• • <i,t%k there is a projection q(ix,..., i,): Ei(X) -> E}(X) for ally and k for which E&X) is defined. (b) E'k(X) is the fibre space induced from the canonical path fibration by Ity.rfft,..., ij-x): Et1 -> TlW-2X(ix,..., i,-x) where the product runs over all (/— l)-tuples, (ilf..., /,_».), such that 1 á »Ï < •• • < ij-i = k. (c) For j<n, B,{x)(tx,..., tj) = h,(l, ...,i,.. .,j) if f, = 0, 1 where, for;>l, AfcO'j.) • • -,i¡): Ek(X) -> P£l'~2X(ix, ■■■, ij) is the composite of projections, Ekk(X)-*££-!_►...-► El +\X)^P&-2X(ix,..., and hfij) is the composite, Ekk(X)-*■-► Definition fibration by ij) F2(Z) -> ÜX(i,). 6.1. (a) E^(X) is the fibre space induced from the canonical path nem_xqdx,..., im-x): ez-\x)-* no-»^,..., im_x) where the product runs over all (m —l)-tuples, (iu ..., i'm_i), such that 1 ^ h < •• < 4,-1 ^ «• (b) The projections, #(/!,..., im): E™(X) -*■E%(X), are induced by the obvious projections from the cartesian product. (c) The universal n-fold commutator, 6n: El(X) -> Q.n~1X, is defined by en(x)(tx,..., t„) = hn(\, ...,f,..., n)(x)(tx, ...Ji,...,tn) forr,=0, 1. One easily verifies that this defines a map E\\(X)^- Q"-1^. Extend the above notation and let E%+1(X) denote the fibre space induced from El(X) by dn. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 340 [April G. J. PORTER Following the notation of [5] let T¡iXy,..., Xn) be the subspace of Xy x - ■■x X„ consisting of those n-tuples with at least ; coordinates at base points. £0 is the cartesian product, Ty is the "fat" wedge and Tn-x(Xx,..., Xn) is the one point union. If Xy= X2= • • • = Xn= X we denote T¿XU ...,Xn)by TRX). Let (/")co be the / skeleton of/" and set p: £*-»■ TSiS1) equal to the quotient map obtained by identifying opposite faces. Clearly p((/")(')) = £^_i(1S'1). Theorem 6.2. (a) £„(!") = A"TS-+i<sl), (b) 6n: EniX)-^- fin_1A" is the map induced by p\ln under the identifications of part (a). Proof, (a) We show that there is a 1-1 correspondence between the points of EniX) and maps T^-^OS1) -»■X. We leave the study of the point set topology to the reader. For all n^2, E&X) = (fiXf and JpS-.i"1»- Xs'"■■■vs\Each of these is in 1-1 correspondence with the set of n-tuples ifi,.. is the composite, .,/„), where fi: S1 -> X. 82 e QX We assume inductively that the theorem is true for E\ when either j<n j=n and i<k. Thus we must show EkiX) = Arr"-* +i(Sl). EkiX) is the subset of EÏ'1 x UPÜk-2Xiiy, ...,ik-y) tuples i<p,TlT¡iix,...,ik-x)) ik-x^n and where the product ijiiu ...,ik-x)\îk~1 or when consisting of (GA)+1)- is taken over all l^i'1<-< = 8k_xqiix,..., ik-x)i<p). By induction <pcorresponds to a map y>:£¿t_fc+2(5'1)-»■X and hence $P: (/«)«-» -+x. For lèix<---<ik-xén, p'^S^x ■■■xS¡k_1) consists of the 2n~k + 1 ik-l) faces of /" on which the coordinates in the (/lf..., 4-i) plane vary and the remainder are constant (0 or 1). We let I^x ■■■xlikl represent these faces. An extension of <pto St\ x ■■■xSf1fc_x<=£nl_fc+1(5'1) is given by an extension of yp to Ih x • ■■xlik_1. Such a map extends fpWiy X *' X Vl) " flfci(»'l,■• -, Ík-l)(<P) and is given by r¡iix,..., /fc_-j)e£fifc_2A'(/1,..., <pto 5,\ x • • • x Silc_1determines ik-y). Conversely, an extension of r,iiy,...,ik_y)eP£lk-2Xiiy,...,ik_y) which restricted to I"'1 is 6kqiix,. ■-, ik-y)i<p). This is true for each ik—l)-tuple 0'i,..., ik-y). Hence maps T£_k+1iS1)^- X are in 1-1 correspondence with the points of EkiX). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1970] HIGHER PRODUCTS Furthermore under this correspondence 341 9n(cp)is the composite F-1-^27(S1)-^>Jf and part (b) follows. A proof similar to the above establishes the theorem when A:= n+1. Corollary 6.3. 0n is null homotopic if and only if the fibration, has a cross-section, (i: T?(Sl) -*■Tg(S1) is inclusion.) Theorem 6.4. If X is an H-space, 6n is null-homotopic. Proof. We construct a map such that i#s~\. s is defined to be the following composite where the first map is induced by the functor Í22, the second is induced by (Ar where r is a retraction SFJtS1) -*■SF^S1), the next map is induced by the adjoint of the identity TS(S^ -> Q.I¡TS(S1), and the last map is induced by a map QEX-»- X such that X-> QS>X-> A1is homotopic to the identity. Such a map exists since X is an //-space. The proof that i#s~ 1 is routine (see the proof of Theorem 2.5 in [5]). Using the universal commutator we now define the analogues of the Massey and Toda products. Definition 6.5. Given <p: Y-*E%(X) define the n-fold commutator, Cn(cp) = <P*[9n]e [Y, ü"-1^]. We say <pis of type (fi,.. .,/„) if <pcovers (fix ■■■xfn)A«: Y->(ÜX)\ (An is the diagonal.) The set of commutators of type (/,,...,/,) Cifix, ■■-,/n) = {Cn(cp)| <pis of type (/,,.. is defined by .,/„)}. These are the commutators arising from the group structure of [ Y, Q.X]. Proposition 6.6. If X is an H space or Y is a suspension then Cn(cp)=0. Proof. If F=Sr and <p:Y-+E&X), there is an adjoint $: Y' -+El(QX) such that the adjoint of Cn(cp)is Cn(<p)e[Y',Q.(ClX)l This reduces the problem to that of the range space being the loop space of an //-space. The proposition is then a corollary to Theorem 6.4. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 342 G. J. PORTER [April To define the higher Samelson products we again need the notion of a special map. Definition 6.7. <p: Yxx ■■■x Yn^~EHX) is special if for each (i'x, ..., iH), 1 á h < ' • • < ik ís « and k < n, there is a map <piix,...,ik): A-tuple Yhx---xYik^Ek+\X) such that the following diagram commutes. <P Yxx---xYn Enn(X) K(k, ■■-,ik) Yhx---xYik <p(ii, - - -, ¡k) Ek + 1(X) -+P£lk^Xiix,...,ik) where the unlabeled maps are canonical projections. Definition 6.8. Given a special map <p: Yxx---x the n-fold Samelson product by Sn(<p)= q(?*lOn]) e[/\(Yx,..., Yn-^E„]iX) define Sn(<p) Yn), fi""1*"] where a is the projection onto a direct summand as in Definition 2.5. We say <p is of type ifi,... ,/„) if <pcovers (/ix---x/n): FiX-.-x Yn-+iQX)\ The set of n-fold Samelson products of type ifi,... ,fi) is defined by (fi, ■■-,fi> = {Sn(<p) | 9 is special of type 0/i,.. .,/„)}. Proposition 6.9. If X is an H-space, Snicp)= 0. Proof. Theorem 6.4. The following naturality properties are easily proven. Theorem 6.10. (a) Ifig: W^ Y, Cni<pg)=g*Cni<p)(b) g*Cnifix,.. .,fin)^Cnifixg,.. .,fng). (c)lfgi-. Wi^Yi,i=l,...,n, (A (gi, ■■-,gn))*sn(<p)= SMgi x • • • xgn)). (d) (A (ft. • • ;gn))*(fl, ■■;fn>c<fgl, - ■-Jgn>- License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1970] 343 HIGHER PRODUCTS Theorem 6.11. Let <fi:X-> U. </.induces El(<f>):El(X)-^E^(U) such that the following diagram commutes for all n. Enn(X)-n-^Çl"-iX EW ii""1^) E*(U)-ÎU-Q""1!/ Given /: QX^-QU it does not necessarily induce a map El(X) -> E£(U) with the above properties. If f(xy)~f(x)f(y), f induces a map E%(X) ->Ff(C/) such that the above diagram homotopy commutes but may not induce a map E3(X) -> El(U). We define the commutative degree of /by setting cd(/)>n if there is an induced map E\\(X) -*■E"(U) such that the diagram given above homotopy commutes. Theorem 6.11 can then be restated as cd (Q/) = co. Consider Q(YX, . ■■, F»)= U?=i CYx x ■■■ x F;x • • • x CYn contained in CYx x ■■■x CYn where CY is the cone on Y. Q(Yx,..., YA is a model for S""1 A (Yx,..., Let w:Q(Yx,..., Yn). Yn)^Tx(ZYx,...,ZYn) he the map which identifies YX<^CY¡to a point for all i. Given cp:TxÇZYx,...,?Yn)-+X, <p*[w]= W(cp) is the nth order Whitehead product studied in [5]. The adjoint map, «„_,: {S""M, B}-*{A, n»-1^} is defined by (an-x(f))(a)(X) =/(A, a) for Ae /». Theorem 6.12. There <p: Yx x ■■■ x Yn->E\\(X) i«n-l)*W(i) is a 1-1 correspondence between special maps, and maps 0: TAJ,Yit . . . ,ZYn)-> X such that = Sn(cp). Proof. By Theorem 6.3 and adjointness there is a 1-1 correspondence between maps <p: YxX--xYn^ E$(X) and maps c}>:Yxx ■■■x Ynx FftS1) -> X. <p determines a map Fi(2 Yx,..., 2 Yn) -* X if and only if <pis special. Thus the first part of the theorem is proven. To prove the last assertion of the theorem let <p: Yrx • • • x rn->- E%(X) be special and let </<:FX(S Ylt..., S Yn)-» X be the map corresponding to <p. <*n-iW(<li)(ti,yx,..., tn,yn) = an-i(<l>w)iti,yi,-.., tn,yn) = 9Íyi,...,yn)(ti,...,tn) = Kiviyu ■■■,yn))ih, ...,tn) = sn(<p)(yx,...,yn)(tu...,tn). We conclude this section by discussing several notions of higher commutativity of a loop space. We first recall that the Whitehead product order of X, WP (X), License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 344 G. J. PORTER [April was defined in [6] to be the largest integer n such that all (n —l)-fold generalized Whitehead products were zero in the homotopy of X. We can also define wp iX) to be the least integer n, such that there are maps/: S T¡ ->■X, i= 1,..., n, which have the property that there is no map 2 Yxx ■■■x 2 Yn -> X of type ifi,.. .,/„). If [fi,.. .,/„] is the set of Whitehead products, Wi<p),where <pis of type (fix,.. .,/„), then WP(A) = n if and only if [fiy,.. .,fn.y] = 0 for all ifi,.. .,fi-x) and there is <p:TyiZYy,...,T,Yn)^ X such that IKOtO^O. wp(A")=n if Oe [fi,.. .,/„_i] for all [/,,.. .,/n_i] and there are gy,...,gn such that 0 £ [glt.. .,gn]. wp iX) = WP (A") if wp (A")< 4. We do not know if this is true in general(4). Let A: SfiA"-^ X be the adjoint of the identity. Proposition 6.13. wp(A") = n ifi and only if 0 e [X,..., X] in-1) 0 $ [X,..., X]n times. times and Proof. Given /: 2 Y-> X there is /: 2 Y^ 2fiA" such that Xf=f. Thus if there exists a map <p:(SfiA")""1 -»■X of type (A)""1, <pO/ix • • • x/n-i) is a map of type (/l,-..,/n-l). Definition 6.14. Define integers SC (Ar), se (A"), WC (A") and wc (A") by (a) SC (A") is the least integer n such that dn is essential. (b) sc (A") is the least integer n such that there exists <p: Yyx ■■• x Yn -> EUX) for some Yu..., Yn with the property thata(<p*[0n])#O. (Note that <pneed not be special.) (c) WC (A") is the least integer n such that there is a special map 9>: YyX-'XYn^EniX) for some spaces Yy,..., Yn with the property that Sni<p)^0. (d) wc (A) is the least integer n such that there are maps fi,...,fin property that 0 £ </,.. with the .,/„>. Proposition 6.15. If X is an H-space SC (A") = sc (A") = wc (A") = WC (A") = oo. Proposition 6.16. SC (X) ^ sc (X) S WC (A")S wc (A"). Proposition 6.17. WC (A")= WP (A); wc (A")= wp (A")(4). It follows from Proposition 6.13 that wc(A")= n if there are liftings <p,: ÍÜ.X)1 ->£j +1(A")of type (1,.. .,1) for j<n but not for j=n. These liftings are given by maps ?j:iQXy^PQ'-1X and hence by maps <p;:(fiA")' xl>: ^- X, j=2,..., n —1. It seems likely that these maps are essentially the maps Q, : (fiX)1 x V'"1 -> fiAr used by Williams [10] to define Cn forms. References 1. H. H. Gershenson, Higher composition products, J. Math Kyoto Univ. 5 (1965), 1-37. MR 32 #8341. (4) Added in proof. F. D. Williams has recently shown that wp(A')=WP(Ar) vtc(X)=WC(X). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use and hence HIGHER PRODUCTS 1970] 345 2. K. A. Hardie, Derived homotopy constructions, J. London Math. Soc. 35 (1960), 465-480. MR 23 #A2883. 3. D. Kraines, Massey higher products, Trans. Amer. Math. Soc. 124 (1966), 431-449. MR 34 #2010. 4. W. S. Massey, Some higher order cohomology operations, Symposium International de Topología Algebraica, Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958, pp. 145-154. MR 20 #4826. 5. G. J. Porter, Higher-order Whiteheadproducts, Topology 3 (1965), 123-135.MR 30 #4261. 6. -, Spaces with vanishing Whitehead products, Quart. J. Math. Oxford Ser. (2) 16 (1965), 77-84. MR 30 #2511. 7. E. Spanier, Higher order operations, Trans. Amer. Math. Soc. 109 (1963), 509-539. MR 28 #1622. 8. J. D. Stasheff, Homotopy associativity of H-spaces. I, Trans. Amer. Math. Soc. 108 (1963), 275-292. MR 28 #1623. 9. H. Toda, Generalized Whitehead products and homotopy groups of spheres, J. Inst. Polytech. Osaka City Univ. Ser. A Math. 3 (1952),43-82. MR 15, 732. 10. F. D. Williams, A characterization of spaces with vanishing generalized higher Whitehead products, Bull. Amer. Math. Soc. 74 (1968),497-499. MR 36 #5944. 11. J. M. Cohen, Thedecompositionof stable homotopy,Ann. of Math. (2) 87 (1968),305-320. MR 37 #6932. 12. J. P. May, Matric Massey products, J. Algebra 12 (1969), 533-568. MR 39 #289. University of Pennsylvania Philadelphia, Pennsylvania 19104 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use