Let $f$ and $g$ are homotopic chain maps of $(C,d)$, then $f_{*n}=g_{*n}: H_{n}(X) \to H_{n}(X')$. Can you give me the example showing that the converse is not true?
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You mean where a collection of maps between homology groups that is not equal to the induced collection from a chain map? – Zelos Malum Jun 21 '16 at 14:01
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I just posted an example here: http://math.stackexchange.com/questions/833135/if-two-chain-maps-over-a-pid-induce-the-same-homomorphism-then-they-are-homotop/1834547#1834547 – iwriteonbananas Jun 21 '16 at 14:24
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Thank you so much :) But I do not know your notation (2 1) mean? @iwriteonbananas – Thế Long Lê Jun 21 '16 at 14:35
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This is the map $\Bbb Z^2\to \Bbb Z$ defined on a basis by $(1,0)\mapsto 2$ and $(0,1)\to 1$ – iwriteonbananas Jun 21 '16 at 15:08
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And (2) mean $x$ becomes $2x$, right? – Thế Long Lê Jun 21 '16 at 15:13