TITLE: Using Mapping cone to show map induce isomorphism on homology
QUESTION [6 upvotes]: 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?

REPLY [5 votes]: 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 $\tilde H_k C(f) = 0$