Multiplication on Spheres (i)

Multiplications exist if n = l, 3, or 7, as follows. In case « = 1, a 1-sphere is formed by the ordinary complex numbers of unit modulus, which have a commutative multiplication. Bott1 [2] has proved that S" does not admit a commutative multiplication if n> 1. A 3-sphere is formed by the quaternions of unit modulus, which have a noncommutative multiplication. H. Samelson [7] and G. Whitehead [12] have proved that the quaternionic multiplication on S3 is not homotopy-commutative. A 7-sphere is formed by the Cayley numbers of unit modulus, which have a noncommutative multiplication. Sugawara [9] has proved that the Cayley multiplication on S7 is not homotopy-commutative. However, there are many classes of multiplications on S3 and S7 besides these, so that (1.1) widens our knowlege quite apart from the possibility of multiplications on spheres of other dimensions. It is an open question whether there are other spheres with multiplications. The obstruction to constructing a multiplication on Sn is the Whitehead product [j,j]Cir2n-i(S''), where/ denotes a generator of irn(Sn). Hence, by (3.72) of [ll], we have: