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A2 N...) A,+, Is a Sequence of Maps, Then {Fi THE DECOMPOSITION OF STABLE HOMOTOPY* BY JOEL M. COHEN UNIVERSITY OF CHICAGO Communicated by Saunders Mac Lane, March 16, 1967 The purpose of this paper is to show that under the operation of higher Toda bracket (which will be defined here) certain stable homotopy classes of spheres, called the Hopf classes, generate all stable homotopy classes of spheres. Further- more, if X is any spectrum (for example, the spectrum SY for any space Y), then the homotopy groups of X (for example, the stable homotopy groups of a space Y) are generated, using these brackets, by the stable homotopy groups of spheres and by those homotopy classes of X not in the kernel of the Hurewicz homomorphism. These results resolve some conjectures about stable homotopy-in particular those raised recently by J. P. May6 in connection with his decomposition of the Adams spectral sequence in terms of matric Massey products. The definition of three-fold bracket is equivalent to that of Toda.9 The defini- tion of higher Toda bracket is believed to be equivalent (at least stably) to those of Spanier7 and Gershenson.4 The details of these results will appear soon. The Bracket Operation.-Conventions: All spaces will have a fixed base point. All maps will be cofibrations in the sense of Spanier.8 (Any map between CW com- plexes is a cofibration.) If two spaces X,Y are of the same homotopy type, we will Write X -- Y. If Y c X, then X/ Y means X with Y identified to the base point. Since all maps are cofibrations, Y C X means that (X, Y) has the homotopy exten- sion property, so there is a canonical map X/Y -- SY. [X, Y] is the set of homotopy classes [f] of maps f X Y. r* (S) is the stable homotopy ring of the sphere. Definition of the set of spaces {fin.. fn} If A1 A2 n...) A,+, is a sequence of maps, then {fi. ..,fn is the set of all spaces C for which there is a filtration An+1 X C X2 C cXn+lC-- such that X,+1/X, - SrAn-r+l and such that the composition STAn-r+l " Xr+ll/Xr * SXr * S(X/Xlr-1) - SrAn-r+2 is homotopic to Srfnr+l. There are induced maps jc An+, - X1 c Xn+1 - C and uc: C X -+l )X+llXn _ SnAl. To include the case n = 0, let { } be the class of topological spaces X with base point. Let jx = v* = identity of X. Definition of the (n + 2)-fold bracket: Given a sequence of maps fo fi .f,+ Ao -' Al -* A2 .. An+1A --An An+2, we define (fo, fl,. ,fn, fn+i) to be the (possibly empty) set of all 6 E [SnAo, An+2] ob- taimed as follows: for some C e If1,... ifn there are maps h: SnAo -* C and 9: C A +2 with Snfo - o- a- h andfn+l g - j, anid 0 = [g - hj. 1175 Downloaded by guest on September 28, 2021 1176 MATHEMATICS: J. M. COHEN PROC. N. A. S. The twofold bracket (Jo, fi) is the unique class [f' o fo]; i.e., we choose C e { where C = range fo = domain fi. r t If f: V S - V S" is a map between wedges of spheres in the stable range j=1 j=1 (i.e., max {nlj < 2 min {mj - 2), then [f] determines and is determined by an (r X t) matrix (([afj])) where asj: S'z -, Smi. Then one can consider the set (Ao, Al,... An, An+,) where Ao is a (1 X ri) row matrix andAiis an (rj X ri+i) matrix all of whose entries are in homotopy groups of spheres, and A,+, an (rn+l X 1) column matrix with entries in the homotopy groups of some space X. Then (Ao,. ..,A.+,) is a (possibly empty) subset of 7rm(X) for some m. Results.-Definition of Hopf classes: Let f: Sm S'. Let Y = S" U rCSm be the mapping cone of f (see ref. 8). Then H"(Y;G) G, Hm+'(Y;G') ' G'. We say that [f] = 0 E 7rm(Sn) is a Hopf class if and only iff is the identity, or for some G, G' there is a primary cohomology operation R of type (G,n;G',m + 1), such that for some x E Hn(Y;G), R(x) 5 O. For such cases, we say that R detects 0. The following is a result of Adams' and Liulevicius5: PROPOSITION 1. There are elements & e 7ro(S), n e ira(S), P e 7r3(S), a e 7r7(S), and al(p) 7r2p-3(S) for odd primes p, and all stable Hopf classes are linear combinations of these. Definition of decomposable: If X is a spectrum and K e 7rm(X), then K is S-decompos- able if there are (ri X ri+i) matrices Ai, i = O... ,n + 1, ro = rn+2 = 1, where A has entries in 7r* (S), i = o,... ,n, A.+, has entries inl 7r* (X) of degree less than m, and K e (Ao,... ,An+). Also, K is decomposable if additionally An+, has entries in positive degree. Our results may now be stated: THEOREM 1. If K E ir* (S) is not a Hopf class, then K is decomposable (i.e., in terms of the bracket operation). THEOREM 2. Let X be a spectrum with K 6 7rm(X) in the kernel of the Hurewicz homomorphism. Then K is S-decomposable. The main tool in this proof is the spectral sequence discussed in reference 3 which was used for some computations in the stable homotopy groups of spheres. We recall that in this spectral sequence we had E*2 - H* (K ;7r* (S)) where K is the Eilenberg-Mac Lane spectrum of Z. Furthermore, E' = 0 for (s,t) # (0,0), El; - Z. The idea of the proof is that E 2 - 7rg(S) but E = 0 for t $ 0. Thus each element in 7r*(S) of positive degree is the boundary, for some dr, of some lower term. If a particular class 0 e ir* (S) - E2 * is the boundary of some term z Tj 0 ,u e H* (K ;ir* (S)) (for ~i E 7r* (S)), then a is in a higher bracket whose last terms is the column matrix of them,. Thus 0 is decomposable if the degree of the Ji is posi- tive. But if the ,Ut are in ro(S), then T = 0i j e H* (K ;iro(S)) - H*(K) and a certain relation exists between T and 0 which implies that T has a dual, a cohomol- ogy operation, which detects 0. Thus 0 is a Hopf class by definition. Theorem 2 is obtained in almost the same way, studying the spectral sequence where E** - H*(K;7r*(X)) and E** gives the quotients in a filtration of H*(X). The edge homomorphism 7r*(X) --,E E- * c H* (X) is the Hurewicz homo- morphism. Thus any element in the kernel of the Hurewicz homomorphism is a boundary, hence is S-decomposable. Downloaded by guest on September 28, 2021 VOL. 57, 1967 MATHEMATICS: J. M. COHEN 1177 By similar methods we can obtain the following: THEOREM 3. Let X and Y be convergent spectra whose homology is finitely generated in each dimension. Let f: X -- Y and g: Y -- X be maps inducing isomorphisms on the image of the Hurewicz homomorphism. Then f and g are weak homotopy equiv- alences. In the study of the cobordism theory2 for a group G (for example G = 0, 'l, Spin, etc.) certain manifolds, called "almost G-manifolds," represent elements in 7r* (MG) where MG is the Thom spectrum of G. The notion of decomposing a manifold as the Toda bracket of lower-dimensional manifolds has a geometric significance. It follows from Theorem 2 that if the cobordism class of a manifold M is in the kernel of the Hurewicz homomorphism, then M is cobordant to a manifold which is so decomposable. * This work was partially supported by NSF grant GP-5609. 1 Adams, J. F., "On the non-existence of elements of Hopf invariant one," Ann. Math., 72, 20- 103 (1960). 2 Brown, E. H., F. P. Peterson, and A. Liulevicius, "Cobordism theories," Ann. Math., 84, 91-101 (1966). 3 Cohen, J. M., "Some results on the stable homotopy groups of spheres," Bull. Am. Math. Soc., 72, 732-735 (1966). 4 Gershenson, H., "Higher composition products," to appear. 5 Liulevicius, A., "The factorization or cyclic reduced powers by secondary operations," Mem. Am. Math. Soc., no. 42 (1962). 6 May, J. P., "The algebraic Eilenberg-Moore spectral sequence," to appear. Spanier, E. H., "Higher order operations", Trans. Am. Math. Soc., 109, 509-539 (1963). 8 Spanier, E. H., Algebraic Topology (New York: McGraw-Hill, 1966). 9 Toda, H., Composition Methods in Homotopy Groups of Spheres (Princeton, N. J.: Princeton University Press, 1962). Downloaded by guest on September 28, 2021.
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