DIVISION ALGEBRAS AND FIBRATIONS OF SPHERES BY GREAT SPHERES

Smooth fibrations of spheres by great spheres occur naturally in the study of the Blaschke conjecture.In fact, if M is a Blaschke manifold, m is a point of M, T m M is the tangent space of M at m, exρ w : T m M -> M is the exponential map at m, and Cut(w) is the cut locus of m in M, then exp^(Cut(m)) is a sphere S m in T m M of center 0, and exp w : S m -» Cut(w) is a smooth great sphere fibration of the sphere S m .For general information of the Blaschke conjecture, see [2].If K is the real, complex, quaternionic or Cayley algebra, n is the dimension of K as a euclidean space, which is 1,2,4 or 8, and S 2n ~ι is the unit (In -l)-sphere in the euclidean 2«-space K X K, then there is a natural smooth great (n -l)-sphere fibration of S 2n ~x such that any (w, w), (u\ w') G S 2 "" 1 belong to the same fibre iff either w = w' = 0 or uw~λ -u'W~λ.When n > 1, this fibration, as well as isomorphic ones, is often referred as the Hopf fibration.Related to this result, Adams' theorem [1] says that a smooth fibration of S 2n ~λ by (n -l)-spheres can occur only when n-1,2,4 or 8, and a classical theorem of Hurwitz [4] says that any division algebra K, which possesses a norm such that for any t>, w E K, | vw | = | v \ | w \ , must be the real, complex, quaternionic or Cayley algebra.If n -1 or 2, then any ^-dimensional division algebra is the real or complex algebra, and any fibration of S 2n ~x by (n -l)-spheres is unique up to an isomorphism.Hence in these cases, the correspondence between ^-dimensional division algebras and smooth great (n -l)-sphere fibrations of S 2 "" 1 is trivial.In this paper, we show that for n -4 or 8, each ^-dimensional division algebra K determines a smooth great (n -l)-sphere fibration of S 2n ~\ and every smooth great (n -l)-sphere fibration of S 2n ~\ up to an isomorphism, is determined by an ^-dimensional division algebra K.However, it is possible