Framing 3-manifolds with bare hands

TL;DR

This paper presents three elementary proofs that all closed, orientable 3-manifolds are parallelizable, avoiding advanced concepts like spin structures and Stiefel-Whitney classes.

Contribution

It introduces simplified proofs for the parallelizability of 3-manifolds that require minimal background knowledge.

Findings

All closed, orientable 3-manifolds are parallelizable.

Three proofs provided without using spin structures or Stiefel-Whitney classes.

Proofs are accessible with minimal mathematical background.

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.