Mapping cylinder

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

The bottom Y is a deformation retract of . The projection splits (via ), and a deformation retraction is given by:

(where points in stay fixed, which is well-defined, because for all ).

The map 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.[3]

Interpretation[edit]

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.

Mapping cylinder.png

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[edit]

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 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 to points of and hence of embedding within despite the fact that the function need not be one-to-one.

Categorical application and interpretation[edit]

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 and .

Mapping telescope[edit]

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[edit]

References[edit]

  1. ^ Hatcher, Allen (2003). Algebraic topology. Cambridge: Cambridge Univ. Pr. p. 2. ISBN 0-521-79540-0. 
  2. ^ Hatcher, Allen (2003). Algebraic topology. Cambridge: Cambridge Univ. Pr. p. 15. ISBN 0-521-79540-0. 
  3. ^ Aguado, Alex. "A Short Note on Mapping Cylinders". arXiv:1206.1277Freely accessible [math.AT].