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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?

1 Answers1

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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 $\tilde H_k C(f) = 0$

Drew
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    Can you tell me about how is the sequence derived (it seems not to be relative homology)? –  Sep 23 '13 at 12:55
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    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 $\tilde H_n(Y,X) = \tilde H_n(Y/X)$ – Drew Sep 23 '13 at 22:40
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    Ok, I'll try to read about cofiberation, I did't know such a concept..is there an recommended book for that? Thanks –  Sep 24 '13 at 09:00
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    @drew Can you expand what you mean here? I'm not quite sure. – Eric Auld Feb 22 '16 at 05:53