Puppe sequence

Remarks

The key step is that the mapping cone of $i \colon Y \to C_f$ is homotopy equivalent to $\Sigma X$, by collapsing the cone on $Y$, which is a contractible subspace attached along a cofibration. Exactness at $[Y, Z]_*$ is direct. A map $g \colon Y \to Z$ extends over $C_f$ exactly when $g \circ f$ is null-homotopic by a pointed homotopy, since such a null-homotopy is a map on $CX$. Repeating the step gives the whole sequence, and up to sign the maps after $q$ are suspensions of the earlier ones. The terms $[\Sigma^n X, Z]_*$ are groups for $n \ge 1$ and abelian groups for $n \ge 2$, and there the exact sequence is one of groups. When $f$ is the inclusion of a subcomplex $X$ of a CW complex $Y$, the space $C_f$ is homotopy equivalent to the quotient $Y / X$. The dual construction with loop spaces and homotopy fibres gives the fibre sequence $\cdots \to \Omega Y \to F \to X \to Y$.

Going deeper

For a student

The Puppe sequence is the source of long exact sequences in homotopy theory. Applied to the inclusion of a subcomplex $A \subseteq X$ and to a representing space $Z$, such as an Eilenberg–MacLane space, it yields the long exact sequence of a pair in cohomology. Its Eckmann–Hilton dual is the fibre sequence, which yields the long exact sequence of homotopy groups of a fibration. It is a basic example of a triangulated structure, which in the stable homotopy category becomes the distinguished triangles $X \to Y \to C_f \to \Sigma X$.