Retraction (topology)

"Retract" redirects here. For other meanings including concepts in group theory and category theory, see Retraction (disambiguation).

In topology, a branch of mathematics, 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

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). 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 I^ω are absolute retracts.

Neighborhood retract

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. The n-sphere Sⁿ is an absolute neighborhood retract.

Deformation retract and strong deformation retract

A continuous map

$F:X\times [0,1]\to X\,$

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

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

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

$F(a,t)=a\,$

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 $(X,A)$ 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.

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 and vice versa. However, there exist contractible spaces which do not strongly deformation retract to a point.

External links

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