Mapping cylinder

In mathematics, specifically algebraic topology, the mapping cylinder of a continuous function formula_1 between topological spaces formula_2 and formula_3 is the quotient
where the formula_5 denotes the disjoint union, and ∼ is the equivalence relation generated by
That is, the mapping cylinder formula_7 is obtained by gluing one end of formula_8 to formula_3 via the map formula_1. Notice that the "top" of the cylinder formula_11 is homeomorphic to formula_2, while the "bottom" is the space formula_13. It is common to write formula_14 for formula_7, and to use the notation formula_16 or formula_17 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 formula_19, 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 formula_7.
The projection formula_21 splits (via formula_22), and the deformation retraction formula_23 is given by:
(where points in formula_26 stay fixed because formula_27 for all formula_28).
The map formula_29 is a homotopy equivalence if and only if the "top" formula_11 is a strong deformation retract of formula_7. An explicit formula for the strong deformation retraction can be worked out.
Examples.
Mapping cylinder of a fiber bundle.
For a fiber bundle formula_32 with fiber formula_33, the mapping cylinder
has the equivalence relation
for formula_36. Then, there is a canonical map sending a point
formula_37 to the point formula_38, giving a fiber bundle
whose fiber is the cone formula_40. To see this, notice the fiber over a point formula_41 is the quotient space
where every point in formula_43 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 formula_44, the mapping cylinder is a space formula_7, "together with" a cofibration formula_46 and a surjective homotopy equivalence formula_21 (indeed, "Y" is a deformation retract of formula_7), such that the composition formula_49 equals "f".
Thus the space "Y" gets replaced with a homotopy equivalent space formula_7, and the map "f" with a lifted map formula_51. 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 formula_54 with the assumption that formula_55 and that formula_1 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 formula_2 to points of formula_58 and hence of embedding formula_2 within formula_58 despite the fact that the function need not be one-to-one.
Categorical application and interpretation.
One can use the mapping cylinder to construct homotopy colimits: this follows from the general statement that any category with all pushouts and coequalizers has all colimits. That is, 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 formula_44 and formula_62.
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 formula_64), 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 end-to-end. The picture of the construction looks like a stack of increasingly large cylinders, like a telescope.
Formally, one defines it as