Given a map $f : X \to Y$, the mapping cone $C(f)$ is the space obtained from the mapping cylinder $M(f)$ by identifying the
subspace $X \times \{0\}$ to a single point.
How can I construct an isomorphism between the homology group $H_n(M(f),X \times \{0\})$ and the reduced homology group $\tilde{H}_n(C(f))$. I can prove they are isomorphic, but how can I construct actually the isomorphism between them? Any help please.
Best Answer
If you take $p$ to be the quotient map $p:(M(f),X \times \{0\})\to(C(f),P)$ (where $P$ is the point $X \times \{0 \}$ is collapsed to) the map this $p$ induces in homology is what I believe you are looking for. It's the only candidate I can think of off the top of my head for a 'natural' choice of homomorphism between the two, indeed it should be an isomorphism. (Note that the reduced homology of $C(f)$ is just the homology of the pair $(C(f),P)$, so that looks reasonable.)