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. Basic properties 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. Examples Mapping cylinder of a fiber bundle 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. 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 , 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. 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