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?
Algebraic Topology – Using Mapping Cone to Show Map Induces Isomorphism on Homology
algebraic-topology
Best Answer
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 $\tilde H_k C(f) = 0$