Puppe sequence

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

(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.

From this, the Puppe sequence gives the homotopy sequence of a fibration:

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.

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.