A note on a short proof of the parallelizability of orientable $3$-manifolds

Abstract

We survey, complete, and modify a proof, involving knot theory, of Stiefel's theorem that all orientable $3$-manifolds are parallelizable. The completion of the proof is done by using the relationship between the tangent bundle and normal bundle of manifolds with non-trivial boundary and on stably parallelizable and parallelizable manifolds. We end with a remark on $7$-manifolds and present J. Korba\v{s}' example of a non-parallelizable $7$-manifold.