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In Hatcher's Algebraic Topology Corollary 3A.7(about p266), he seemed to used a fact that if a map whose reduced homology of the mapping cone are all zero , then it induces isomorphism on the homology. Can anyone help me to understand this?

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For any map $f:X \to Y$ there is an associated long exact sequence in reduced homology $$ \cdots \to \tilde H_n(X) \to \tilde H_{n}(Y) \to \tilde H_{n}(C(f)) \to \tilde H_{n-1} (X) \to \cdots $$

Your result then follows from the fact that $\tilde H_k C(f) = 0$

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    $\begingroup$ Can you tell me about how is the sequence derived (it seems not to be relative homology)? $\endgroup$ – Qixiao Sep 23 '13 at 12:55
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    $\begingroup$ It is in disguise. Up to homotopy we can assume $f$ is a cofibration, and so we can identify $C(f)$ with $Y/X$, and then this is just the long exact sequence in (relative) homology along with an identification $\tilde H_n(Y,X) = \tilde H_n(Y/X)$ $\endgroup$ – Drew Sep 23 '13 at 22:40
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    $\begingroup$ Ok, I'll try to read about cofiberation, I did't know such a concept..is there an recommended book for that? Thanks $\endgroup$ – Qixiao Sep 24 '13 at 9:00
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    $\begingroup$ @drew Can you expand what you mean here? I'm not quite sure. $\endgroup$ – Eric Auld Feb 22 '16 at 5:53

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