# Deformation retract

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In topology, a branch of mathematics, a **retraction**,^{[1]} 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

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.

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*. 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 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

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.)

## Neighborhood deformation retract

A pair of spaces in U is an NDR-pair if there exists a map such that and a homotopy such that for all , for all , and for all . The pair is said to be a representation of as an NDR-pair.

## Properties

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. Contractibility, however, is a weaker condition, as contractible spaces exist which do not deformation retract to a point.^{[2]}

## Notes

## External links

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

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