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

.

be a continuous map between pointed spaces 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. 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:

1. (the) various constructions (of the coexact sequence) involve quotient spaces instead of subspaces, and so all maps and homotopies require more scrutiny to ensure that they are well-defined and continuous.

Examples

Example: Relative homotopy

As a special case, one may take X to be a subspace A of Y that contains the basepoint y0, and f of A into Y. One then obtains an exact sequence in the category of pointed spaces:

are the homotopy groups denotes the homotopy equivalence of maps from U to W. One may then show that

is in bijection, thus giving rise to the relative homotopy sequence of pairs

.

Example: Fibration

As a special case, one may take f to be a fibration. Then the mapping fiber Mp has the homotopy lifting property and it follows that Mp 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, that there is a bijection

1. .

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

be a continuous map between CW complexes denote a mapping cone of f, (i.e., the cofiber of the map f), so that we have a (cofiber) sequence:

1. .

suspensions of A and B (this is because suspension might be seen as a functor), obtaining a sequence:

1. .

Note that suspension preserves cofiber sequences.

is homotopy equivalent Thus we have a sequence:

:

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:

1. .

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, 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 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).

References

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