Bott Periodicity and the Parallelizability of the spheres

Abstract

The theorems of Bott (4), (5) on the stable homotopy of the classical groups imply that the sphere Sn is not parallelizable for n ≠ 1, 3, 7. This was shown independently by Kervaire(8) and Milnor(7), (9). Another proof can be found in (3), §26·11. The work of J. F. Adams (on the non-existence of elements of Hopf invariant one) implies more strongly that Sn with any (perhaps extraordinary) differentiable structure is not parallelizable if n ≠ 1, 3, 7. Thus there exist already four proofs for the non-parallelizability of the spheres, the first three mentioned relying on the Bott theory, as given in (4), (5). The purpose of this note is to show how the refined form of Bott's results given in (6) leads to a very simple proof of the non-parallelizability (only for the usual differentiable structures of the spheres). We shall prove in fact the following theorem due to Milnor (9) which implies the non-parallelizability.