I can't figure out how to prove the following:
Let $M$ be an $n$-manifold. Then
$\bigwedge^nT^∗M$ is trivial $\Leftrightarrow M$ is orientable
Thank you
2026-03-26 20:39:49.1774557589
$\bigwedge^nT^∗M$ is trivial $\Leftrightarrow M$ is orientable
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The bundle is one-dimensional, and such a thing is trivial iff it has a nonzero section. Now a nonzero section of that bundle is precisely what is known as a volume form, and a manifold is orientable iff it has a volume form.