Elementary proof of the fact that any orientable 3-manifold is parallelizable 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?

Kirby gives a nice proof that every orientable 3-manifold is parallelizable at https://math.berkeley.edu/~kirby/papers/Kirby%20-%20The%20topology%20of%204-manifolds%20-%20MR1001966.pdf . (See the unnumbered non-blank page between page 46 and page 47.) 
