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Wikipedia / Mapping cylinder

In mathematics, specifically algebraic topology, the mapping cylinder of a continuous function between topological spaces is the quotient

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

is homeomorphic 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 is given by:

).

is a homotopy equivalence. An explicit formula for the strong deformation retraction can be worked out.

Examples

Mapping cylinder of a fiber bundle

For a fiber bundle, the mapping cylinder

has the equivalence relation

, giving a fiber bundle

is the quotient space

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:

, together with and a surjective homotopy equivalence (indeed, Y is a deformation retract), such that the composition equals f.

Thus the space Y, and the map f. 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 is actually the inclusion of a subspace.

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 pushoutss and coequalizers has all colimitss. 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.

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 end-to-end. The picture of the construction looks like a stack of increasingly large cylinders, like a telescope.

Formally, one defines it as

See also

References