The reduced homology of a complement in $S^n$ is the shifted reduced cohomology of the subset.Suggest a slogan
(Alexander duality)
Let \(n\geq 1\) and let \(K\subsetneq S^n\) be a nonempty, compact, locally contractible subspace. Then for all \(i\), \[ \tilde H_i(S^n \sm K; \ZZ) \cong \tilde H^{n-i-1}(K;\ZZ) .\]
It follows, with \(M = S^n\) and the long exact sequence of the pair \((S^n, S^n\sm K)\), from the duality \(H_i(M, M\sm K;\ZZ)\cong H^{n-i}(K;\ZZ)\) for a compact, locally contractible subspace \(K\) of a closed, orientable \(n\)-manifold \(M\) [@Hat02].
[@Hat02].