This question is somehow related to the generalized Jordan curve theorem. I have already showed that if $h:S^k\to S^n$ is an embedding ($0\leq k<n$), then $\tilde{H}_i(S^n\setminus h(S^k))\cong\mathbb{Z}$ if $i=n-k-1$ and $\cong0$ otherwise.
Then I need to prove the theorem for $\mathbb{R}^n$ (identify it as $S^n$ minus a point), using Mayer-Vietoris sequence, I get
$$\tilde{H}_{i+1}(S^n\setminus h(S^k))\to\tilde{H}_{i+1}(S^n)\to\tilde{H}_{i+1}(\mathbb{R}^n\setminus h(S^k))\to\tilde{H}_i(S^n\setminus h(S^k))\to\tilde{H}_i(S^n)$$
Then if $k>0$, we get $\tilde{H}_i(\mathbb{R}^n\setminus h(S^k))\cong\mathbb{Z}$ if $i=n-k-1$ or $i=n-1$ and $\cong0$ otherwise. For $k=0$, we get $\tilde{H}_i(\mathbb{R}^n\setminus h(S^0))\cong0$ if $i\neq n-1$, but if $i=n-1$, I only get a long exact sequence. So what is $\tilde{H}_{n-1}(\mathbb{R}^n\setminus h(S^0))$ essentially? (for $n=1,2$, this is $\mathbb{Z}^2$ using geometric properties. I guess this is also true for all $n$, but how to show this?) Am I missing something here? Certainly this theorem is not as concise as the one with $S^n$.