In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deformation retraction is a mapping that captures the idea of continuously shrinking a space into a subspace. An absolute neighborhood retract (ANR) is a particularly well-behaved type of topological space. For example, every topological manifold is an ANR. Every ANR has the homotopy type of a very simple topological space, a CW complex.

Definitions

Retract

Let X be a topological space and A a subspace of X. Then a continuous map 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 the inclusion, a retraction is a continuous map r such that 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 (any constant map yields a retraction). If X is Hausdorff, then A must be a closed subset of X. If r: X \to A is a retraction, then the composition ι∘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.

Deformation retract and strong deformation retract

A continuous map is a deformation retraction of a space X onto a subspace A if, for every x in X and a in 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 a homotopy equivalence. A retract need not be a deformation retract. For instance, having a single point as a deformation retract of a space X would imply that X is path connected (and in fact that X is contractible). Note: An equivalent definition of deformation retraction is the following. A continuous map r: X \to 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 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 Hatcher, take this as the definition of deformation retraction.) As an example, the n-sphere S^{n} is a strong deformation retract of as strong deformation retraction one can choose the map Note that the condition of being a strong deformation retract is strictly stronger than being a deformation retract. For instance, let X be the subspace of consisting of closed line segments connecting the origin and the point (1/n, 1) for n a positive integer, together with the closed line segment connecting the origin with (0,1). Let X have the subspace topology inherited from the Euclidean topology on. Now let A be the subspace of X consisting of the line segment connecting the origin with (0,1). Then A is a deformation retract of X but not a strong deformation retract of X.

Cofibration and neighborhood deformation retract

A map f: A → X of topological spaces is a (Hurewicz) cofibration if it has the homotopy extension property for maps to any space. This is one of the central concepts of homotopy theory. A cofibration f is always injective, in fact a homeomorphism to its image. If X is Hausdorff (or a compactly generated weak Hausdorff space), then the image of a cofibration f is closed in X. Among all closed inclusions, cofibrations can be characterized as follows. The inclusion of a closed subspace A in a space X is a cofibration if and only if A is a neighborhood deformation retract of X, meaning that there is a continuous map with and a homotopy such that H(x,0) = x for all x \in X, H(a,t) = a for all a \in A and and if u(x) < 1. For example, the inclusion of a subcomplex in a CW complex is a cofibration.

Properties

No-retraction theorem

The boundary of the n-dimensional ball, that is, the (n−1)-sphere, is not a retract of the ball. (See .)

Absolute neighborhood retract (ANR)

A closed subset X of a topological space Y is called a neighborhood retract of Y if X is a retract of some open subset of Y that contains X. Let \mathcal{C} be a class of topological spaces, closed under homeomorphisms and passage to closed subsets. Following Borsuk (starting in 1931), a space X is called an absolute retract for the class \mathcal{C}, written if X is in \mathcal{C} and whenever X is a closed subset of a space Y in \mathcal{C}, X is a retract of Y. A space X is an absolute neighborhood retract for the class \mathcal{C}, written if X is in \mathcal{C} and whenever X is a closed subset of a space Y in \mathcal{C}, X is a neighborhood retract of Y. Various classes \mathcal{C} such as normal spaces have been considered in this definition, but the class \mathcal{M} of metrizable spaces has been found to give the most satisfactory theory. For that reason, the notations AR and ANR by themselves are used in this article to mean and. A metrizable space is an AR if and only if it is contractible and an ANR. By Dugundji, every locally convex metrizable topological vector space V is an AR; more generally, every nonempty convex subset of such a vector space V is an AR. For example, any normed vector space (complete or not) is an AR. More concretely, Euclidean space \reals^{n}, the unit cube I^{n},and the Hilbert cube I^{\omega} are ARs. ANRs form a remarkable class of "well-behaved" topological spaces. Among their properties are: