Let $M$ be a smooth manifold (not necessarily orientable) and let $N=\partial M$. Is $N$ necessarily orientable?
I have no particular reason to believe that this is the case, but I wasn't able to come up with a counterexample either.
2026-05-14 06:18:06.1778739486
Boundaries of Manifolds Necessarily Orientable
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I'm a bit rusty on this but would not something like $M\times [0,1)$ be a counterexample? (where $M$ is the Möbius band)