Puppe sequence

In mathematics, the Puppe sequence is a construction of homotopy theory. It comes in two forms: an long exact sequence, built from the mapping fibre (a fibration), and a long coexact sequence, built from the mapping cone (which is a cofibration). Intuitively, the Puppe sequence allows us to think of homology theory as a functor that takes spaces to long-exact sequences of groups. It is also useful as a tool to build long exact sequences of relative homotopy groups.

Coexact Puppe sequence

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) → ....

Exact Puppe sequence

Let $f:(X,x_{0})\to (Y,y_{0})$ be a continuous map between pointed spaces and let $Mf$ denote the mapping fibre (the fibration dual to the mapping cone). One then obtains an exact sequence:

$Mf\to X\to Y$

where the mapping fibre is defined as:

$Mf=\{(x,\omega )\in X\times Y^{I}:\omega (0)=y_{0}{\mbox{ and }}\omega (1)=f(x)\}$

Observe that the loop space $\Omega Y$ injects into the mapping fibre: $\Omega Y\to Mf$, as it consists of those maps that both start and end at the basepoint $y_{0}$. One may then show that the above sequence extends to the longer sequence

$\Omega X\to \Omega Y\to Mf\to X\to Y$

The construction can then be iterated to obtain the exact Puppe sequence

$\cdots \to \Omega ^{2}(Mf)\to \Omega ^{2}X\to \Omega ^{2}Y\to \Omega (Mf)\to \Omega X\to \Omega Y\to Mf\to X\to Y$

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).

References

• E. Spanier, Algebraic Topology, Springer-Verlag (1982) Reprint, McGraw Hill (1966)