Complement homology of a compact codimension-zero submanifold (${\tilde{H}}_i(S^n\setminus W;\mathbb{Z})\cong {\tilde{H}}^{n-i-1}(W;\mathbb{Z})$)

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.