Framing 3-manifolds with bare hands

Abstract

After surveying existing proofs that every closed, orientable 3-manifold is parallelizable, we give three proofs using minimal background. In particular, our proofs use neither spin structures nor the theory of Stiefel-Whitney classes.

15 pages. Expanded and more complete version. We survey a further proof available in the literature based on even surgery presentations of 3-manifolds and we provide two further "bare hands" proofs based on the existence of such presentations in a few special cases