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.
Let be a continuous map between pointed spaces and let 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 injects into the mapping fibre: , as it consists of those maps that both start and end at the basepoint . 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:
(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.
As a special case, one may take X to be a subspace A of Y that contains the basepoint y0, and f to be the inclusion of A into Y. One then obtains an exact sequence in the :
where the are the homotopy groups, is the zero-sphere (i.e. two points) and denotes the homotopy equivalence of maps from U to W. Note that . One may then show that
is in bijection to the relative homotopy group , thus giving rise to the relative homotopy sequence of pairs
The object is a group for and is abelian for .
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 and the fiber have the same homotopy type.
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In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber) is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces . It acts as a homotopy theoretic kernel of a mapping of topological spaces due to the fact it yields a long exact sequence of homotopy groupsMoreover, the homotopy fiber can be found in other contexts, such as homological algebra, where the distinguished trianglegives a long exact sequence analogous to the long exact sequence of homotopy groups.
In mathematics, especially homotopy theory, the mapping cone is a construction of topology, analogous to a quotient space. It is also called the homotopy cofiber, and also notated . Its dual, a fibration, is called the mapping fibre. The mapping cone can be understood to be a mapping cylinder , with one end of the cylinder collapsed to a point. Thus, mapping cones are frequently applied in the homotopy theory of pointed spaces. Given a map , the mapping cone is defined to be the quotient space of the mapping cylinder with respect to the equivalence relation , .
In mathematics, in particular homotopy theory, a continuous mapping between topological spaces is a cofibration if it has the homotopy extension property with respect to all topological spaces . That is, is a cofibration if for each topological space , and for any continuous maps and with , for any homotopy from to , there is a continuous map and a homotopy from to such that for all and . (Here, denotes the unit interval .
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