Retract

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

$r:X \to A$

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

$r \circ \iota = \operatorname{id}_A,$

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 non-empty 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 $r:X \to A$ is a retraction, then the composition $\iota \circ r$ is an idempotent continuous map from X to X. Conversely, given any idempotent continuous map $s:X\to X$, 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 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 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

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

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 Sⁿ is a strong deformation retract of Rⁿ⁺¹\{0}; as strong deformation retraction one can choose the map

$F(x,t)=\left((1-t)+{t\over \|x\|}\right) x.$

Neighborhood deformation retract

A closed subspace A is a neighborhood deformation retract of X if there exists a continuous map $u:X \rightarrow I$ (where $I=[0,1]$) such that $A = u^{-1} (0)$ and a homotopy $H:X\times I\rightarrow X$ such that $H(x,0) = x$ for all $x \in X$, $H(a,t) = a$ for all $(a,t) \in A\times I$, and $h(x,1) \in A$ for all $x \in u^{-1} [ 0 , 1)$.^[2]

Properties

• The main obvious property of a retract A of X is that any continuous map $f : A \rightarrow Y$ has at least one extension $g : X \rightarrow Y$, namely, $g=f\circ r\,$.
• 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]

No-retraction theorem

The boundary of the n-dimensional ball, that is, the (n − 1)-sphere, is not a retract of the ball, which is known as the no-retraction theorem. (See Brouwer fixed-point theorem#A proof using homology.)

Notes

1. K. Borsuk (1931). "Sur les rétractes". Fund. Math. 17: 2–20.
2. Steenrod, N. E. (1967). "A convenient category of topological spaces". Michigan Math. J. 14 (2): 133–152. doi:10.1307/mmj/1028999711.
3. Hatcher, Allen (2002), Algebraic topology, Cambridge University Press, ISBN 978-0-521-79540-1

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.