= Complement homology of a compact codimension-zero submanifold
{title2=$\widetilde H_i(S^n\setminus W;\mathbb Z)\cong\widetilde H^{n-i-1}(W;\mathbb Z)$}
A compact smooth $n$-manifold with nonempty boundary embedded in $S^n$ is a proper compact locally contractible subset. <Alexander duality> therefore expresses the <reduced homology> of its complement by the reduced <cohomology> of the abstract manifold. Thus these groups do not depend on its particular <smooth embedding>.
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