Deformation retract

"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 ^[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

$r:X \to A$

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.

Notes

1. ^ K. Borsuk (1931). "Sur les rétractes". Fund. Math. 17: 2–20. 

References

• Hatcher, Allen (2002), Algebraic topology, Cambridge University Press, ISBN 978-0-521-79540-1, http://www.math.cornell.edu/~hatcher/AT/ATpage.html 

External links

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