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.

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


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

is a deformation retraction 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 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: X &rarr; 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.