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In topology, a retraction, as the name suggests, "retracts" an entire space into a subspace. A deformation retraction is a map which captures the idea of continuously shrinking a space into a subspace.
## Definitions

### Retract

### Neighborhood retract

### Deformation retract and Strong deformation retract

## Neighborhood deformation retract

## Properties

## References

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

- $r:X\; to\; A$

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

- $iota\; :\; A\; hookrightarrow\; X$

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

- $r\; circ\; iota\; =\; id\_A,$

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).

A space X is known as an absolute retract (or AR) if for every normal space Y that embeds X as a closed subset, X is a retract of Y.

If there exists an open set U such that

- $A\; subset\; U\; subset\; X$

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.

A continuous map

- $d:X\; times\; [0,\; 1]\; to\; X$

is a deformation retraction if, for every x in X and a in A,

- $d(x,0)\; =\; x,\; ;\; d(x,1)\; in\; A\; ,quad\; mbox\{and\}\; quad\; d(a,1)\; =\; a.$

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

- $d(a,t)\; =\; a,$

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

A pair $(X,\; A)$ of spaces in U is an NDR-pair if there exists a map $u:X\; rightarrow\; I$ such that $A\; =\; u^\{-1\}\; (0)$ and a homotopy $h:I\; times\; X\; rightarrow\; X$ such that $h(0,\; x)\; =\; x$ for all $x\; in\; X$, $h(t,\; a)\; =\; a$ for all $(t,\; a)\; in\; I\; times\; A$, and $h(1,\; x)\; in\; A$ for all $x\; in\; u^\{-1\}\; [0\; ,\; 1)$. The pair $(h,\; u)$ is said to be a representation of $(X,\; A)$ as an NDR-pair.

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.

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Last updated on Monday September 22, 2008 at 21:05:24 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Monday September 22, 2008 at 21:05:24 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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