Toward a Global Understanding of the Homotopy Groups of Spheres

In this paper we will describe a point of view that has emerged as a result of research on the homotopy groups of spheres in the last decade. This philosophy is difficult to translate into theorems or even into precise conjectures, and it is certainly not apparent in the formal literature on the subject. With the exception of Theorem 10, we will not present any proofs or announcements of new results here. Rather we will collect numerous old results and current ideas and arrange them into what we hope is a suggestive picture. 1. General facts about homotopy groups For the last 50 years one of the basic problems in algebraic topology has been the determination of the homotopy groups of spheres πn+k(S), i.e. the classification of continuous maps S → S up to continuous deformation. The simplicity of the spaces involved lends intuitive appeal to the problem, but experience has shown that it is as hard as any in mathematics. There have been several major computational breakthroughs in the subject, namely the EHP sequence (to be described in Section 7 below), and the spectral sequences of Serre, Adams and Novikov. Each of these had lead to a large amount of new information but has also increased our appreciation of the difficulty of the problem. Much of this material is dealt with in greater depth and with numerous references in [R1]. We begin by recalling some of the basic facts about the problem. All of these groups are abelian and finitely generated. The groups πn+k(S) are known to vanish when k < 0 and when n = 1 and k > 0. The group πn(S) is isomorphic to the integers Z. These were all proved by Hurewicz around 1935. Their finite computability was established by E.H. Brown in 1959. The following finiteness result was proved by Serre. Theorem 1. [S]. The groups πn+k(S) for k > 0 are all finite with the exception of π4n−1(S), which is the direct sum of Z and a finite abelian group. Hence for n odd the standard map S → K(Z, n)