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If two complexes have isomorphic integral homology, do the dual complexes have isomorphic integral cohomology? I can also assume that the homology, cohomology are finitely-generated abelian groups.

The question I really care about is the following. If two simply-connected spaces (manifolds) have isomorphic integral homology, does that imply that they have isomorphic integral cohomology? Clearly this is true over fields by the universal coefficient theorem. Note that I am not assuming that the isomorphism is induced by a map of spaces.

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Yes, this is true over $\mathbb{Z}$ by the universal coefficient theorem over $\mathbb{Z}$.

For the question with "homology" and "cohomology" switched see this math.SE question.

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