On the Hopf-Toda invariant

Abstract

0. Introduction. One line of attack on the problem of computing the unstable homotopy groups of spheres is to attempt to construct the elements whose reduced product filtration exceeds 1. The classical Hopf construction associates with a map S' x St1 S of type (oc,,B) an element of itm+t(S'+). The indeterminacy of the construction is the suspension subgroup in the sense that the elements associated with any two maps of the same type differ by a suspension. However, it was proved by I. M. James [8] that the filtration of the element obtained does not exceed 2. In [3] I defined a generalization of the Hopf construction which can yield elements of arbitrarily large filtration. Thus, for example, let a denote the homotopy boundary homomorphism and let Ox E 7rr(Sr'2), fi ;(SL1, eSr r2) be elements(1) such that the Whitehead product [cc,afl] is trivial. Then there exists a map F: S'm X St1 * S-2 of type (, af). Applying the construction we obtain an element C(F) e7E i+t(Sn+ 1) with indeterminacy equal to the subgroup of filtration r 2. As always the problem is to prove nontriviality. To this end we study a homomorphism