PARALLELIZABLE MANIFOLDS AND HOMOTOPY GROUPS OF SPHERES

<正> The main results in this paper are: for a connected parallelizable closed differentiablemanifold M~n, if n = 3 or 7, then we have （i） all tangent framings on M produce all eithereven or odd elements in π_ns, which are even or odd depending on the Euler semi-characteris-tic X*（M） mod 2 being 0 or 1; （ii） for a fixed imbedding of M~n in R2n+1, the set of elementsin π2n+1(Sn+1) produced by tangent framings and the set of elements with even [odd] Hopfinvariant are the same, if X*（M）≡0[≡1] mod 2.