Induced mapping into mapping cylinder

Question

Can anyone please tell what is the induced map from mapping cone of f into Z? Also please explain how H and g induced G.

Thanks in advance

Answer 1

A map $f: X \to Y$ is nullhomotopic iff there is an extension to $\tilde{f}:CX \to Y$ - you can just define $\tilde{f}\left(x, t\right) = h_t\left(x\right)$ where $h_t$ is the nullhomotopy.

$C_f$ can be constructed as the pushout $CX \cup_f Y$, so by the universal property of pushout, mapping out of it is the same as mapping from $CX \sqcup Y$ in a way that that agrees on the image of $f$. A map from a disjoint union is the same as a pair of maps from each of the components. $H$ is then the extension of the composition to the cone, and $g$ is the given map, and they agree on $f$, thus define a map from the mapping cone.