A short and elementary proof that a product of spheres is parallelizable if one of them is odd

Abstract

Kervaire [1] has shown that the product of several spheres, one of which is of odd dimension, is a parallelizable manifold. This note observes a short and elementary proof of this fact. The question does not arise if the spheres are all even dimensional, for the Euler characteristic shows that there is not even a nonzero vector field. First observe that the tangent bundle of a product manifold, r(M1XM2), is canonically isomorphic to the Whitney sum p 1'rM1 Ep1rM2 where p1rMi is the bundle induced by pi: M1XM2->Mi (the projection) from rM, (i = 1, 2). Secondly, if M1, M2 are s-parallelizable (i.e. AEDrMi_imi+l where t is a trivial line bundle and im is the m-fold Whitney sum of t with itself) then M1 XM2 is s-parallelizable. This follows from: