Puppe Sequence

In mathematics, the Puppe sequence is a construction of homotopy theory.

Let f:A → B be a continuous map between CW complexes and let C(f) denote a cone of f, so that we have a sequence:

A → B → C(f).

Now we can form ΣA and ΣB, suspensions of A and B respectively, and also Σf: ΣA → ΣB (this is because suspension might be seen as a functor), obtaining a sequence:

ΣA → ΣB → C(Σf).

Now one notices quite easily, that C(Σf) is homotopy equivalent to ΣC(f) and that one have a natural map C(f) → ΣA (this is defined, roughly speaking, by collapsing B ⊆ C(f) to a point). Thus we have a sequence:

A → B → C(f) → ΣA → ΣB → ΣC(f).

Iterating this construction, we obtain the Puppe sequence associated to A → B:

A → B → C(f) → ΣA → ΣB → ΣC(f) → Σ2A → Σ2B → Σ2C(f) → Σ3A → Σ3B → Σ3C(f) → ....