Theorem $S^{n-1}$ disconnects $S^n$ into two open connected components, which have $S^{n-1}$ as frontier.

In $R^3$, if we replace sphere of standard torus with genus $g\geq1$, we may have "The Jordan-Brouwer Separation Theorem" intuitively. Then what happens when we replace topological sphere of topology torus with genus $g\geq1$?

 

We have $T^n$ in $R^{n+1}$, so do we have the same theorem?

 

What is the situation ahout manifold?

Theorem $H_k(S^n-S^r)=R,k=n-r-1$.

Does this theorem have the corresponding generalization?

Theorem $S^{n-1}$ disconnects $S^n$ into two open connected components, which have $S^{n-1}$ as frontier.

In $R^3$, if we replace sphere of standard torus with genus $g\geq1$, we may have "The Jordan-Brouwer Separation Theorem" intuitively. Then what happens when we replace topological sphere of topology torus with genus $g\geq1$?

We have $T^n$ in $R^{n+1}$, so do we have the same theorem?

What is the situation ahout manifold?

Theorem $H_k(S^n-S^r)=R,k=n-r-1$.

Does this theorem have the corresponding generalization?