TITLE: Elementary proof of the fact that any orientable 3-manifold is parallelizable
QUESTION [34 upvotes]: A parallelizable manifold $M$ is a smooth manifold such that there exist smooth vector fields $V_1,...,V_n$ where $n$ is the dimension of $M$, such that at any point $p\in M$, the tangent vectors $V_1(p),...,V_n(p)$ provide a basis for the tangent space at $p$. Equivalently, a manifold is parallelizable if its tangent bundle is trivial.
There is a theorem that states that any compact orientable 3-manifold is parallelizable, and 
there is a proof of this result which uses $spin^c$ structures and the Steifel-Whitney class.
I am wondering whether there exists a more elementary, perhaps more straightforward proof. Otherwise, I would be grateful for some intuition on why this is true.
Also, it the theorem still true without the compactness assumption? If so, is there a relatively simple proof in that case?

REPLY [4 votes]: Benedetti and Lisca wrote a paper on this same question. They write “the answers given [on this question] until July 18, 2018 use
the same tools employed in the proofs mentioned above”, and give three more-elementary answers.
https://arxiv.org/abs/1806.04991