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.


1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.