I want to prove the generalized Jordan Curve Theorem via homology theory. In order to do this I have to show that $\widetilde{H}_k(S^n\setminus f(\mathbb{D}^m))=0$ for all $k$, where $\mathbb{D}^m$ is the $m$-disk and $f$ is an embedding. I would prove this last claim by induction and by Mayer-Vietoris sequence, whitouth us the concept of limit. Are there ideas or references?
$\widetilde{H}_k(S^n\setminus f(\mathbb{D}^m))=0$
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algebraic-topology
homology-cohomology