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Concept
Mapping cylinder
Summary
In mathematics, specifically algebraic topology, the mapping cylinder of a continuous function between topological spaces and is the quotient
where the denotes the disjoint union, and ∼ is the equivalence relation generated by
That is, the mapping cylinder is obtained by gluing one end of to via the map . Notice that the "top" of the cylinder is homeomorphic to , while the "bottom" is the space . It is common to write for , and to use the notation or for the mapping cylinder construction. That is, one writes
with the subscripted cup symbol denoting the equivalence. The mapping cylinder is commonly used to construct the mapping cone , obtained by collapsing one end of the cylinder to a point. Mapping cylinders are central to the definition of cofibrations.
The bottom Y is a deformation retract of .
The projection splits (via ), and the deformation retraction is given by:
(where points in stay fixed because for all ).
The map is a homotopy equivalence if and only if the "top" is a strong deformation retract of . An explicit formula for the strong deformation retraction can be worked out.
For a fiber bundle with fiber , the mapping cylinder
has the equivalence relation
for . Then, there is a canonical map sending a point
to the point , giving a fiber bundle
whose fiber is the cone . To see this, notice the fiber over a point is the quotient space
where every point in is equivalent.
The mapping cylinder may be viewed as a way to replace an arbitrary map by an equivalent cofibration, in the following sense:
Given a map , the mapping cylinder is a space , together with a cofibration and a surjective homotopy equivalence (indeed, Y is a deformation retract of ), such that the composition equals f.
Thus the space Y gets replaced with a homotopy equivalent space , and the map f with a lifted map . Equivalently, the diagram
gets replaced with a diagram
together with a homotopy equivalence between them.
The construction serves to replace any map of topological spaces by a homotopy equivalent cofibration.
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Mapping cone (topology)
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 , .
Homotopy fiber
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.