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

Dionne Ibarra

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arXiv · 2207.12149

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

Dionne Ibarra

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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.

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Dionne Ibarra. 2022-07-25. A note on a short proof of the parallelizability of orientable $3$-manifolds. https://arxiv.org/abs/2207.12149

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