BERNOULLI NUMBERS , HOMOTOPY GROUPS , A N D A THEOREM OF ROHLIN By JOHN W . MILNOR AND MICHEL

A homomorphism J: 7Tk_1(SOw) -> nm+k_1(S ) from the homotopy groups of rotation groups to the homotopy groups of spheres has been defined by H. Hopf and G. W. Whitehead. This homomorphism plays an important role in the study of differentiable manifolds. We will study its relation to one particular problem: the question of possible Pontrjagin numbers of an 'almost parallelizable' manifold. Definition. A connected differentiable manifold M with base point XQ is almost parallelizable if M — xQ is parallelizable. If M k is imbedded in a high-dimensional Euclidean space B (m ^ k+1) then this is equivalent to the condition that the normal bundle v, restricted to M — x0, be trivial (compare the argument given by Whitehead , or Kervaire P , §8)). The following theorem was proved by Rohlin in 1952 (see Rohlin', Kervaire). Theorem (Rohlin). Let Jf be a compact oriented differentiable ̂ -manifold with Stiefel-Whitney class w2 equal to zero. Then the Pontrjagin number p-^M] is divisible by 48. Rohlin's proof may be sketched as follows. It may be assumed that M is a connected manifold imbedded in B, m ^ 5. Step 1. It is shown that Jf is almost parallelizable. Let / b e a cross-section of the normal SOm-bundle v restricted to Jf — xQ. The obstruction to extending/is an element