Mapping cylinder

Mapping cylinder

In mathematics, specifically algebraic topology, the mapping cylinder of a function f between topological spaces X and Y is the quotient

$M_f = (([0,1]\times X) \amalg Y)\,/\,\sim$

where the union is disjoint, and ∼ is the equivalence relation

$(0,x)\sim f(x)\quad\text{for each }x\in X.$

That is, the mapping cylinder Mf is obtained by gluing one end of X × [0,1] to Y via the map f. Notice that the "top" of the cylinder $\{1\}\times X$ is homeomorphic to X, while the "bottom" is the space Y.

See [1] for more details.

Basic properties

The bottom Y is a deformation retract of Mf. The projection $M_f \to Y$ splits (via $Y \ni y \mapsto y \in Y \subset M_f$), and a deformation retraction R is given by:

$R: M_f \times I \rightarrow M_f$
$([t,x],s) \mapsto [s\cdot t,x]$

(where points in Y stay fixed, which is well-defined, because $[0,x]=[s\cdot 0,x]$ for all s).

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 $f\colon X \to Y$, the mapping cylinder is a space Mf, together with a cofibration $\tilde f\colon X \to M_f$ and a surjective homotopy equivalence $M_f \to Y$ (indeed, Y is a deformation retract of Mf), such that the composition $X \to M_f \to Y$ equals f.

Thus the space Y gets replaced with a homotopy equivalent space Mf, and the map f with a lifted map $\tilde f$. Equivalently, the diagram

$f\colon X \to Y$

gets replaced with a diagram

$\tilde f\colon X \to M_f$

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 $f\colon X\rightarrow Y$ with the assumption that $X \subset Y$ and that f 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 X to points of Y, and hence of embedding X within Y, 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 limits[citation needed]: 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 $f\colon X \to Y$ and $\text{id}_X\colon X \to X$.

Mapping telescope

Given a sequence of maps

$X_1 \to_{f_1} X_2 \to_{f_2} X_3 \to\cdots$

the mapping telescope is the homotopical direct limit. If the maps are all already cofibrations (such as for the orthogonal groups $O(n) \subset O(n+1)$), 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

$\Bigl(\coprod_i [0,1] \times X_i\Bigr) / ((0,x_i) \sim (1,f(x_i)))$

References

1. ^ Algebraic Topology by Allen Hatcher. Page 2

• Mapping cylinder (homological algebra)

Wikimedia Foundation. 2010.

Look at other dictionaries:

• Mapping cone — In mathematics, especially homotopy theory, the mapping cone is a construction Cf of topology, analogous to a quotient space. It is also called the homotopy cofiber, and also notated Cf. Contents 1 Definition 1.1 Example of circle …   Wikipedia

• Mapping cone (homological algebra) — In homological algebra, the mapping cone is a construction on a map of chain complexes inspired by the analogous construction in topology. In the theory of triangulated categories it is a kind of combined kernel and cokernel: if the chain… …   Wikipedia

• Cylinder-head-sector — Cylinder head sector, also known as CHS, was an early method for giving addresses to each physical block of data on a hard disk drive. In the case of floppy drives, for which the same exact diskette medium can be truly low level formatted to… …   Wikipedia

• List of mathematics articles (M) — NOTOC M M estimator M group M matrix M separation M set M. C. Escher s legacy M. Riesz extension theorem M/M/1 model Maass wave form Mac Lane s planarity criterion Macaulay brackets Macbeath surface MacCormack method Macdonald polynomial Machin… …   Wikipedia

• List of algebraic topology topics — This is a list of algebraic topology topics, by Wikipedia page. See also: topology glossary List of topology topics List of general topology topics List of geometric topology topics Publications in topology Topological property Contents 1… …   Wikipedia

• Cofibration — In mathematics, in particular homotopy theory, a continuous mapping , where A and X are topological spaces, is a cofibration if it satisfies the homotopy extension property with respect to all spaces Y. The name is because the dual condition, the …   Wikipedia

• Cylindre d'application —  Pour l’article homonyme, voir Cylindre (homonymie).  où où …   Wikipédia en Français