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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

2011
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Bulletin of the American Mathematical Society
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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

doi:10.1090/s0273-0979-2011-01345-3
fatcat:pn5oj3vt5vcfrkno4qsrdne2mm