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. 



Does there exists a (possibly homological) characterization of the Jordan curve property in all dimensions?
More genreally, the number (finite or not) of the connected components of the complement set of a compact subset $M\subset {\mathbb R}^n$, which is the rank of $H\_0({\mathbb R}^n\setminus M)$, is a homotopic invariant for compact subspaces of ${\mathbb R}^n$, by duality in homology. 


