A Note on Topological Parallelizability

A differentiate manifold M is said to be parallelizable if the tangent vector bundle of M is trivial. A topological manifold M is said to be topologically parallelizable if the tangent microbundle of M is trivial. In [2] Milnor has shown that on some open set M in some Euclidean space Rn there exists a differentiable structure with respect to which the integral Pontrjagin class p(M) of M is different from 1. It follows that on a topologically parallelizable manifold it is possible to have a differentiable structure with respect to which the manifold is not parallelizable. It is known that the only spheres (of dimension 2:1) which are differentiably parallelizable are S1, S3 and S7 [l]. It is also known that the only spheres which have fibre homotopically trivial tangent sphere bundles are S1, S3 and S7 [$]. In this note we prove