Definitions

Deformation retract

In topology, 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).

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.

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.

Deformation retract and Strong deformation retract

A continuous map

$d:X times \left[0, 1\right] to X$

is a deformation retraction if, for every x in X and a in A,

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

In other words, a deformation retraction is a homotopy between a retract 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

$d\left(a,t\right) = a,$

for all t in [0, 1], d is called a strong deformation retraction. In other words, a strong deformation retraction leaves points in A fixed throughout the homotopy. (Some authors take this as the definition of deformation retraction.)

Neighborhood deformation retract

A pair $\left(X, A\right)$ of spaces in U is an NDR-pair if there exists a map $u:X rightarrow I$ such that $A = u^\left\{-1\right\} \left(0\right)$ and a homotopy $h:I times X rightarrow X$ such that $h\left(0, x\right) = x$ for all $x in X$, $h\left(t, a\right) = a$ for all $\left(t, a\right) in I times A$, and $h\left(1, x\right) in A$ for all $x in u^\left\{-1\right\} \left[0 , 1\right)$. The pair $\left(h, u\right)$ is said to be a representation of $\left(X, A\right)$ 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.

References

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