Mapping cylinder

Summary

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

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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  .[2] An explicit formula for the strong deformation retraction can be worked out.[3]

Examples

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Mapping cylinder of a fiber bundle

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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

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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

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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

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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

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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

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References

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  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:1206.1277 [math.AT].
  • May, J.P (1999). A Concise Course in Algebraic Topology. The University of Chicago Press. ISBN 978-0-2265-1183-2.