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

TL;DR

This paper reviews and completes a proof demonstrating that all orientable 3-manifolds are parallelizable, using knot theory and bundle relationships, and discusses implications for higher-dimensional manifolds.

Contribution

It provides a complete, modified proof of the parallelizability of orientable 3-manifolds, connecting knot theory with bundle theory and extending insights to 7-manifolds.

Findings

All orientable 3-manifolds are parallelizable.

The proof involves knot theory and bundle relationships.

There exist non-parallelizable 7-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.