Deformation retract

From formulasearchengine
Revision as of 14:17, 28 May 2012 by en>Chricho (interwiki, please notice that some of the linked articles cover more than just deformation retracts)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search

{{#invoke:Hatnote|hatnote}}Template:Main other 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.



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 In as well as the Hilbert cube Iω are absolute retracts.

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 Sn 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: XA 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.


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]


  1. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
  2. {{#invoke:citation/CS1|citation |CitationClass=citation }}

External links

de:Retraktion und Koretraktion fi:Deformaatioretrakti it:Retrazione pl:Retrakt deformacyjny ru:Деформационный ретракт zh:形变收缩