THE DECOMPOSITION OF STABLE HOMOTOPY

Abstract

The purpose of this paper is to show that, under the operation of higher Toda bracket (which will be defined in ? 2), certain classes in the stable homotopy ring of spheres w*(S) generate all of w*(S). These classes, called the Hopf classes, are those detected by primary cohomology operations. We show further that if X is any spectrum, then w*(X) is generated by w*(S) acting by higher Toda brackets on those elements of w*(X) not in the kernel of the Hurewicz homomorphism. The proofs provide on algorithm for finding such brackets. These results answer conjectures raised recently by J. P. May [11] in connection with his decomposition of the E2-term of the Adams spectral sequence into Massey higher products on certain terms. Our definition of 3-fold Toda bracket is equivalent to Toda's toral construction [15]. The definition of the higher brackets is believed to be equivalent (at least in the stable range) to those of Spanier [13]. (Our notation is the reverse of these authors, but is more in keeping with that of Massey products.) The author wishes to thank Professor A. Liulevicius and Professor B. Gray for many fruitful disscussions. Furthermore, the author expresses his appreciation of the clarifying suggestions of both the reference and the editor. The results of this paper have been announced in [6].