Mapping cylinder

In mathematics, specifically algebraic topology, the mapping cylinder[1] of a continuous function

denotes the disjoint union, and ~ is the equivalence relation generated by That is, the mapping cylinder

is obtained by gluing one end of

Notice that the "top" of the cylinder

That is, one writes with the subscripted cup symbol denoting the equivalence.

, 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

is a homotopy equivalence if and only if the "top"

is a strong deformation retract of

[2] An explicit formula for the strong deformation retraction can be worked out.

, the mapping cylinder has the equivalence relation for

Then, there is a canonical map sending a point

To see this, notice the fiber over a point

is the quotient space where every point in

, the mapping cylinder is a space

and a surjective homotopy equivalence

The construction serves to replace any map of topological spaces by a homotopy equivalent cofibration.

Note that pointwise, a cofibration is a closed inclusion.

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

Another, more intuitive appeal of the construction is that it accords with the usual mental image of a function as "sending" points of

despite the fact that the function need not be one-to-one.

One can use the mapping cylinder to construct homotopy colimits:[citation needed] 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

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 picture of the construction looks like a stack of increasingly large cylinders, like a telescope.