# Difference between revisions of "Deformation retract"

(removing some idiocy) |
|||

(One intermediate revision by one other user not shown) | |||

Line 1: | Line 1: | ||

{{redirect|Retract|other meanings including concepts in group theory and category theory|Retraction (disambiguation)}} | {{redirect|Retract|other meanings including concepts in group theory and category theory|Retraction (disambiguation)}} | ||

− | In [[topology]], a branch of mathematics, a '''retraction''' | + | In [[topology]], a branch of mathematics, a '''retraction'''<ref>{{cite journal|title=Sur les rétractes|author=K. Borsuk|journal=Fund. Math.|volume=17|year=1931|pages=2–20}}</ref> is a continuous mapping from the entire [[space (mathematics)|space]] into a [[Subspace topology|subspace]] which preserves the position of all points in that subspace. A '''deformation retraction''' is a [[function (mathematics)|map]] which captures the idea of ''[[continuous function|continuously]] shrinking'' a space into a subspace. |

== Definitions == | == Definitions == | ||

Line 50: | Line 50: | ||

:<math>F(a,t) = a\,</math> | :<math>F(a,t) = a\,</math> | ||

− | for all ''t'' in [0, 1], ''F'' is called a '''strong deformation retraction'''. In other words, a strong deformation retraction leaves points in ''A'' fixed throughout the homotopy. (Some authors, such as [[Allen Hatcher]], take this as the definition of deformation retraction.) | + | for all ''t'' in [0, 1] and ''a'' in ''A'', then ''F'' is called a '''strong deformation retraction'''. In other words, a strong deformation retraction leaves points in ''A'' fixed throughout the homotopy. (Some authors, such as [[Allen Hatcher]], take this as the definition of deformation retraction.) |

As an example, the [[n-sphere|''n''-sphere]] ''S<sup>n</sup>'' is a strong deformation retract of '''R'''<sup>''n''+1</sup>\{0}; as strong deformation retraction one can choose the map | As an example, the [[n-sphere|''n''-sphere]] ''S<sup>n</sup>'' is a strong deformation retract of '''R'''<sup>''n''+1</sup>\{0}; as strong deformation retraction one can choose the map | ||

Line 58: | Line 58: | ||

A closed subspace ''A'' is a '''neighborhood deformation retract''' of ''X'' if there exists a continuous map <math>u:X \rightarrow I</math> (where <math>I=[0,1]</math>) such that <math>A = u^{-1} (0)</math> and a homotopy | A closed subspace ''A'' is a '''neighborhood deformation retract''' of ''X'' if there exists a continuous map <math>u:X \rightarrow I</math> (where <math>I=[0,1]</math>) such that <math>A = u^{-1} (0)</math> and a homotopy | ||

<math>H:X\times I\rightarrow X</math> such that <math>H(x,0) = x</math> for all <math>x \in X</math>, <math>H(a,t) = a</math> for all | <math>H:X\times I\rightarrow X</math> such that <math>H(x,0) = x</math> for all <math>x \in X</math>, <math>H(a,t) = a</math> for all | ||

− | <math>(a,t) \in A\times I</math>, and <math>h(x,1) \in A</math> for all <math>x \in u^{-1} [ 0 , 1)</math>.<ref name='steenrod'>{{cite journal | journal= Michigan Math. J. | last1=Steenrod | first1=N. E. | title=A convenient category of topological spaces | volume=14 | issue=2 | year=1967 | pages=133–152}}</ref> | + | <math>(a,t) \in A\times I</math>, and <math>h(x,1) \in A</math> for all <math>x \in u^{-1} [ 0 , 1)</math>.<ref name='steenrod'>{{cite journal | journal= Michigan Math. J. | last1=Steenrod | first1=N. E. | title=A convenient category of topological spaces | volume=14 | issue=2 | year=1967 | pages=133–152 | doi=10.1307/mmj/1028999711}}</ref> |

==Properties== | ==Properties== | ||

− | + | * The main obvious property of a retract ''A'' of ''X'' is that ''any'' continuous map <math>f : A \rightarrow Y</math> has at least one ''extension'' <math>g : X \rightarrow Y</math>, namely, <math>g=f\circ r\,</math>. | |

− | Any topological space which deformation retracts to a point is [[contractible space|contractible]] and vice versa. However, there exist contractible spaces which do not strongly deformation retract to a point.<ref name='hatcher'>{{Citation | last1=Hatcher | first1=Allen | title=Algebraic topology | url=http://www.math.cornell.edu/~hatcher/AT/ATpage.html | publisher=[[Cambridge University Press]] | isbn=978-0-521-79540-1 | year=2002}}</ref> | + | * Deformation retraction is a particular case of homotopy equivalence. In fact, two spaces are [[homotopy equivalent]] [[if and only if]] they are both deformation retracts of a single larger space. |

+ | |||

+ | * Any topological space which deformation retracts to a point is [[contractible space|contractible]] and vice versa. However, there exist contractible spaces which do not strongly deformation retract to a point.<ref name='hatcher'>{{Citation | last1=Hatcher | first1=Allen | title=Algebraic topology | url=http://www.math.cornell.edu/~hatcher/AT/ATpage.html | publisher=[[Cambridge University Press]] | isbn=978-0-521-79540-1 | year=2002}}</ref> | ||

==Notes== | ==Notes== | ||

{{Reflist}} | {{Reflist}} | ||

+ | |||

+ | == References == | ||

+ | *J.P. May, A concise course in algebraic topology | ||

+ | *[[James Munkres|Munkres, James]]; ''Topology'', Prentice Hall; 2nd edition (December 28, 1999). ISBN 0-13-181629-2. | ||

==External links== | ==External links== |

## Latest revision as of 03:27, 30 May 2014

{{#invoke:Hatnote|hatnote}}Template:Main other
In topology, a branch of mathematics, a **retraction**^{[1]} is a continuous mapping from the entire space into a subspace which preserves the position of all points in that subspace. A **deformation retraction** is a map which captures the idea of *continuously shrinking* a space into a subspace.

## Definitions

### Retract

Let *X* be a topological space and *A* a subspace of *X*. Then a continuous map

is a **retraction** if the restriction of *r* to *A* is the identity map on *A*; that is, *r*(*a*) = *a* for all *a* in *A*. Equivalently, denoting by

the inclusion, a retraction is a continuous map *r* such that

that is, the composition of *r* with the inclusion is the identity of *A*. Note that, by definition, a retraction maps *X* onto *A*. A subspace *A* is called a **retract** of *X* if such a retraction exists. For instance, any space retracts to a point in the obvious way (the constant map yields a retraction). If *X* is Hausdorff, then *A* must be closed.

If is a retraction, then the composition is an idempotent continuous map from *X* to *X*. Conversely, given any idempotent continuous map , we obtain a retraction onto the image of *s* by restricting the codomain.

A space *X* is known as an **absolute retract** if for every normal space *Y* that contains *X* as a closed subspace, *X* is a retract of *Y*. The unit cube *I ^{n}* as well as the Hilbert cube

*I*are absolute retracts.

^{ω}### Neighborhood retract

If there exists an open set *U* such that

and *A* is a retract of *U*, then *A* is called a **neighborhood retract** of *X*.

A space *X* is an **absolute neighborhood retract** (or **ANR**) if for every normal space *Y* that embeds *X* as a closed subset, *X* is a neighborhood retract of *Y*. The *n*-sphere *S ^{n}* is an absolute neighborhood retract.

### Deformation retract and strong deformation retract

A continuous map

is a *deformation retraction* of a space *X* onto a subspace *A* if, for every *x* in *X* and *a* in *A*,

In other words, a deformation retraction is a homotopy between a retraction and the identity map on *X*. The subspace *A* is called a **deformation retract** of *X*. A deformation retraction is a special case of homotopy equivalence.

A retract need not be a deformation retract. For instance, having a single point as a deformation retract would imply a space is path connected (in fact, it would imply contractibility of the space).

*Note:* An equivalent definition of deformation retraction is the following. A continuous map *r*: *X* → *A* is a deformation retraction if it is a retraction and its composition with the inclusion is homotopic to the identity map on *X*. In this formulation, a deformation retraction carries with it a homotopy between the identity map on *X* and itself.

If, in the definition of a deformation retraction, we add the requirement that

for all *t* in [0, 1] and *a* in *A*, then *F* is called a **strong deformation retraction**. In other words, a strong deformation retraction leaves points in *A* fixed throughout the homotopy. (Some authors, such as Allen Hatcher, take this as the definition of deformation retraction.)

As an example, the *n*-sphere *S ^{n}* is a strong deformation retract of

**R**

^{n+1}\{0}; as strong deformation retraction one can choose the map

### Neighborhood deformation retract

A closed subspace *A* is a **neighborhood deformation retract** of *X* if there exists a continuous map (where ) such that and a homotopy
such that for all , for all
, and for all .^{[2]}

## Properties

- The main obvious property of a retract
*A*of*X*is that*any*continuous map has at least one*extension*, namely, .

- Deformation retraction is a particular case of homotopy equivalence. In fact, two spaces are homotopy equivalent if and only if they are both deformation retracts of a single larger space.

- Any topological space which deformation retracts to a point is contractible and vice versa. However, there exist contractible spaces which do not strongly deformation retract to a point.
^{[3]}

## Notes

## References

- J.P. May, A concise course in algebraic topology
- Munkres, James;
*Topology*, Prentice Hall; 2nd edition (December 28, 1999). ISBN 0-13-181629-2.

## External links

*This article incorporates material from Neighborhood retract on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*