# Using Mapping cone to show map induce isomorphism on homology

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?

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 $\tilde H_k C(f) = 0$
• 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