Puppe sequence

In mathematics, the Puppe sequence is a construction of homotopy theory. Intuitively, the Puppe sequence allows us to think of homology theory as a functor that takes spaces to long-exact sequences of groups.

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

Note that suspension preserves cofiber sequences.

Due to this powerful fact we know that C(Σf) is homotopy equivalent to ΣC(f). By collapsing B ⊆ C(f) to a point, one has a natural map C(f) → ΣA. 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) → ....

Some properties and consequences

It is a simple exercise in topology to see that every three elements of a Puppe sequence are, up to a homotopy, of the form:

X → Y → C(f).

By "up to a homotopy", we mean here that every 3 elements in a Puppe sequence are of the above form if regarded as objects and morphisms in suitable category: homotopy category.

If one is now given a topological half-exact functor, the above property implies that after acting with the functor in question on the Puppe sequence associated to A → B, one obtains a long exact sequence. Most notably this is the case with a family of functors of homology – the resulting long exact sequence is called the sequence of a pair (A,B) (see Eilenberg–Steenrod axioms; However, a different approach is taken in that article and a sequence of a pair is treated there as an axiom).

Remarks

As there are two "kinds" of suspension, unreduced and reduced, one can also consider unreduced and reduced Puppe sequences (at least if dealing with pointed spaces, when it's possible to form reduced suspension).