Is an $(n-1)$-dimensional submanifold of an $n$-dimensional manifold orientable?

Question

Is it true that every $(n-1)$-dimensional submanifold of a compact (compact without boundary) $n$-dimensional manifold is orientable ? I think, that the answer is no, but I can’t provide it with an example. I thought about Möbius strip and Klein bottle. But the dimensions are not corresponding (the difference is 2 , not 1)

Answer 1

If you take $M$ a non-orientable compact $(n-1)$-manifold, then it is a submanifold of $M\times S^1$ which is compact.

For example you can take $M$ to be the Klein bottle.

Answer 2

All the examples in Adam Chalumeau's answer are non-orientable hypersurfaces of non-orientable manifolds. It is worth noting that there are non-orientable hypersurfaces of orientable manifolds. For example, $\mathbb{RP}^{2n}$ is a hypersurface in $\mathbb{RP}^{2n+1}$; the former is non-orientable while the latter is orientable.