# Submanifolds of Orientable Manifolds With Boundary

Let $(M, \partial M)$ be an orientable $n$-dimensional topological manifold with boundary. Suppose that $(N, \partial N)$ is an $n$-dimensional topological manifold with boundary and $N \subset M$.

If $N- \partial N$ is an open subset of $M$ then I believe that $N$ is also orientable since the orientation of $M$ (which can be viewed compatible choice of local orientations at each point $x \in M - \partial M$) should induce an orientation of $N$.

If $N - \partial N$ is not an open subset of $M$ then can I say if $N$ is orientable or not?

• If $\partial M$ is isotopic to $\partial N$, certainly. Mar 14 '13 at 21:14