Commentary on "On the parallelizability of the spheres" by R. Bott and J. Milnor and "On the nonexistence of elements of Hopf invariant one" by J. F. Adams
Bulletin of the American Mathematical Society
In the words of Heine, "Es ist eine alte Geschichte, doch bleibt sie immer neu"an old story which stays forever new. Our old/new story concerns the finite dimensional division algebras over the reals R and real vector bundles over the spheres S n . By definition, an n-dimensional division algebra is an n-dimensional real vector space V with a bilinear multiplication V × V → V without zero divisors; the multiplication need be neither commutative nor associative. The story starts with the purely
... ts with the purely algebraic 19th century theorems of Frobenius and Hurwitz. They proved that there are only four normed division algebras, namely R itself with n = 1, the complex numbers C with n = 2, the quaternions H with n = 4, and the octonions O with n = 8. In 1935 Hopf [Ho] used topology to prove that if there exists an n-dimensional division algebra, then n = 2 k for some k 0. For n 2 an n-dimensional division algebra V determines: 1. A non-trivial n-dimensional vector bundle η(V ) over S n , called the Hopf bundle of V ; 2. A trivialization of the tangent bundle τ S n−1 of S n−1 , so that S n−1 is parallelizable. The Hopf invariant H(f ) ∈ Z of a map f : S 2n−1 → S n showed that the Hopf bundles η(V ) for n = 2, 4, 8 are nontrivial, since they determine maps f with H(f ) = 1. The nontriviality of η(V ) also follows from the nontriviality of the mod 2 Stiefel-Whitney class w n (η(V )) = 0.