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The Alexander duality Theorem is usually stated for a triangulable pair $(\mathbb S^n, Y)$ where $Y$ is a subset of the standard sphere $\mathbb S^n$. My question is: Does the duality also hold if we rather replace $\mathbb S^n$ by a compact orientable Homology sphere (https://en.m.wikipedia.org/wiki/Homology_sphere) ? I'm mainly interested in the cases $n=2$ and $3$.

Thanks

uno
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    What proofs of Alexander Duality do you know? It is usually deduced from Poincare-Lefshetz duality theorem. Incidentally, homology spheres (say, over ${\mathbb Z}$) are automatically compact and orientable. – Moishe Kohan May 23 '20 at 04:33

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