If $S^{2n+1}$ is covering space of $X$, then $X$ is orientable.

Question

Is there any direct way to prove that $n$-manifold is orientable? In AT we can just calculate $n$'th homology group and check whether it's $\mathbb Z$ or $0$. But I want a geometric method, using differential forms. Thanks!

Answer 1

If you have a nowhere vanishing n-form, then the manifold is orientable.