A consequence of Alexander duality is the fact that an $n$-dimensional simplicial complex $K$ embedded in $\mathbb{R}^{n+1}$ separates $\mathbb{R}^{n+1}$ into $\beta_n + 1$ connected components where $\beta_n$ is the dimension of $H_n(K)$.
When $K$ is a 1-dimensional complex embedded in $\mathbb{R}^2$, $K$ is a planar graph. It is well known that the edges in a planar embedding belong to at most two faces of the embedding. That is, the 1-simplices in $K$ belong to the boundaries of at most two connected components of $\mathbb{R}^2 \setminus K$.
Does the statement for planar graphs hold in higher dimensions? If $K$ is an $n$-dimensional complex embedded in $\mathbb{R}^{n+1}$ is it true that the $n$-simplices of $K$ belong to the boundaries of at most two connected components of $\mathbb{R}^{n+1} \setminus K$?