# Deformation retract

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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.*