Puppe sequence

In mathematics, the Puppe sequence is a construction of homotopy theory, so named after Dieter Puppe. It comes in two forms: a 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.
Exact Puppe sequence.
Let formula_1 be a continuous map between pointed spaces and let formula_2 denote the mapping fibre (the fibration dual to the mapping cone). One then obtains an exact sequence:
where the mapping fibre is defined as:
Observe that the loop space formula_5 injects into the mapping fibre: formula_6, as it consists of those maps that both start and end at the basepoint formula_7. One may then show that the above sequence extends to the longer sequence
The construction can then be iterated to obtain the exact Puppe sequence
The exact sequence is often more convenient than the coexact sequence in practical applications, as Joseph J. Rotman explains:
Examples.
Example: Relative homotopy.
As a special case, one may take "X" to be a subspace "A" of "Y" that contains the basepoint "y"0, and "f" to be the inclusion formula_10 of "A" into "Y". One then obtains an exact sequence in the category of pointed spaces:
where the formula_12 are the homotopy groups, formula_13 is the zero-sphere (i.e. two points) and formula_14 denotes the homotopy equivalence of maps from "U" to "W". Note that formula_15. One may then show that 
is in bijection to the relative homotopy group formula_17, thus giving rise to the relative homotopy sequence of pairs
The object formula_19 is a group for formula_20 and is abelian for formula_21.
Example: Fibration.
As a special case, one may take "f" to be a fibration formula_22. Then the mapping fiber "Mp" has the homotopy lifting property and it follows that "Mp" and the fiber formula_23 have the same homotopy type. It follows trivially that maps of the sphere into "Mp" are homotopic to maps of the sphere to "F", that is, 
From this, the Puppe sequence gives the homotopy sequence of a fibration:
Example: Weak fibration.
Weak fibrations are strictly weaker than fibrations, however, the main result above still holds, although the proof must be altered. The key observation, due to Jean-Pierre Serre, is that, given a weak fibration formula_26, and the fiber at the basepoint given by formula_23, that there is a bijection
This bijection can be used in the relative homotopy sequence above, to obtain the homotopy sequence of a weak fibration, having the same form as the fibration sequence, although with a different connecting map.
Coexact Puppe sequence.
Let formula_29 be a continuous map between CW complexes and let formula_30 denote a mapping cone of "f", (i.e., the cofiber of the map "f"), so that we have a (cofiber) sequence:
Now we can form formula_32 and formula_33 suspensions of "A" and "B" respectively, and also formula_34 (this is because suspension might be seen as a functor), obtaining a sequence:
Note that suspension preserves cofiber sequences.
Due to this powerful fact we know that formula_36 is homotopy equivalent to formula_37 By collapsing formula_38 to a point, one has a natural map formula_39 Thus we have a sequence:
Iterating this construction, we obtain the Puppe sequence associated to formula_41:
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:
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 the 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 formula_41, one obtains a long exact sequence.
A result, due to John Milnor, is that if one takes the Eilenberg–Steenrod axioms for homology theory, and replaces excision by the exact sequence of a weak fibration of pairs, then one gets the homotopy analogy of the Eilenberg–Steenrod theorem: there exists a unique sequence of functors formula_45 with "P" the category of all pointed pairs of topological spaces.
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).