On The Homotopy Groups of Spheres and Rotation Groups

Abstract

One of the outstanding problems in modern topology is that of classifying the mappings of an m-dimensional sphere Sm into a topological space X. In terms of the Hurewicz theory of homotopy groups2 this problem may be phrased as follows: to determine the structure of the mth homotopy group 7r.(X). Of particular interest is the case where X itself is an n-sphere S'. In this case the results of Hopf,3 Freudenthal,4 and Pontrjagin5 have led to the solution of the problem for m _ n + 2. For m > n + 2 almost nothing is known concerning the structure of 7rm(S'). That this problem is closely related to the study of homotopy properties of the rotation group Rn of the n-sphere has been shown by Pontrjagin,6 who has used the oneand two-dimensional homotopy groups of Rn to compute the groups Tn+i(S.) (i = 1, 2). In the present paper we introduce an operation which associates with each mapping f(Sm X Sn) C S' a mapping fgm(S+n+1) C Sn+,. This is a generalization of the procedure of Hopf6 for the case m = n. This operation is shown to induce a homomorphism of 7rm(Rn) into Wtm+n+1(Sn+1), which for m = 1, 2 turns out to be an isomorphism. The connection of this homomorphism with one introduced by Freudenthal4 is studied. In a recent paper Freudenthal7 has announced without proof a very general theorem on extension of mappings, and used this theorem to construct maps of S2n-1 on Sn of Hopf invariant 16 for all even n. We shall use the above results to construct a counter-example to Freudenthal's theorem. It is further shown that Freudenthal's construction definitely fails if n > 2 and n 2 (mod 4).