More precisely, let $M$ be a subspace $\mathbb R^n$ with the following properties:

- $M$ is a topological manifold of dimension $n-1$.
- M is compact.

Does there exist a homological characterization of when the following happens:

- $\mathbb R^n \backslash M$ has two components, the bounded one being "inside" and the other one "outside". Both are $n$-dimensional manifolds.

If the above is not possible, is there a different formulation of the question which would allow a nice characterization?

The motivation of this question is of course the realization that the solution for $n = 3$ seems to be that $M$ is an oriented surface.