Deformation retract

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 [0, 1] to X

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

d(x,0) = x, ; d(x,1) in A ,quad mbox{and} quad d(a,1) = 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(a,t) = 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 (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.

References

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