 Mapping cylinder

In mathematics, specifically algebraic topology, the mapping cylinder of a function f between topological spaces X and Y is the quotient
where the union is disjoint, and ∼ is the equivalence relation
That is, the mapping cylinder M_{f} is obtained by gluing one end of X × [0,1] to Y via the map f. Notice that the "top" of the cylinder is homeomorphic to X, while the "bottom" is the space Y.
See ^{[1]} for more details.
Contents
Basic properties
The bottom Y is a deformation retract of M_{f}. The projection splits (via ), and a deformation retraction R is given by:
(where points in Y stay fixed, which is welldefined, because for all s).
Interpretation
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 M_{f}, together with a cofibration and a surjective homotopy equivalence (indeed, Y is a deformation retract of M_{f}), such that the composition equals f.
Thus the space Y gets replaced with a homotopy equivalent space M_{f}, 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.
Note that pointwise, a cofibration is a closed inclusion.
Applications
Mapping cylinders are quite common homotopical tools. One use of mapping cylinders is to apply theorems concerning inclusions of spaces to general maps, which might not be injective.
Consequently, theorems or techniques (such as homology, cohomology or homotopy theory) which are only dependent on the homotopy class of spaces and maps involved may be applied to with the assumption that and that f is actually the inclusion of a subspace.
Another, more intuitive appeal of the construction is that it accords with the usual mental image of a function as "sending" points of X to points of Y, and hence of embedding X within Y, despite the fact that the function need not be onetoone.
Categorical application and interpretation
One can use the mapping cylinder to construct homotopy limits^{[citation needed]}: given a diagram, replace the maps by cofibrations (using the mapping cylinder) and then take the ordinary pointwise limit (one must take a bit more care, but mapping cylinders are a component).
Conversely, the mapping cylinder is the homotopy pushout of the diagram where and .
Mapping telescope
Given a sequence of maps
the mapping telescope is the homotopical direct limit. If the maps are all already cofibrations (such as for the orthogonal groups ), then the direct limit is the union, but in general one must use the mapping telescope. The mapping telescope is a sequence of mapping cylinders, joined endtoend. The picture of the construction looks like a stack of increasingly large cylinders, like a telescope.
Formally, one defines it as
References
 ^ Algebraic Topology by Allen Hatcher. Page 2
See also
 Mapping cylinder (homological algebra)
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