If we know that the homology groups of two chain complex are isomorphic to each other, can we say that there is a chain map between these two chain complexes？ If so, how can we define the chain map?
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There is always the zero map of complexes. Do you mean must there exist a homotopy equivalence between the two complexes, that is, a chain map inducing isomorphisms on the homology groups? – Jared Apr 25 '13 at 05:13
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@Jared: homotopy equivalence is a strictly stronger condition than quasi-isomorphism. – Qiaochu Yuan Apr 25 '13 at 06:47
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Yes, you're right Qiaochu. I should have said quasi-isomorphism. – Jared Apr 25 '13 at 13:39