Abstract. The multiplication problem for spheres is to determine which spheres in Euclidean space Sn-1 -› En permit a continuous multiplication. This paper presents the topological K-theory proof that it is only possible when n = 1, 2, 4, and 8. These cases correspond to S0 -› E1, S1 -› E2, S3 -› E4, and S7 -› E8 where the multiplications are given respectively by the real numbers, complex numbers, quaternions, and Cayley numbers.
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