Some Properties and Consequences
It is a simple exercise in topology to see that every 3 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).
Read more about this topic: Puppe Sequence
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