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 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, 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 XontoA. 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. If X is Hausdorff, then A must be a closed subset of X. If is a retraction, then the composition ι∘r is an idempotent continuous map from X to X. Conversely, given any idempotent continuous map 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. Note: An equivalent definition of deformation retraction is the following. A continuous map 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 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. As an example, the n-sphere is a strong deformation retract of as strong deformation retraction one can choose the map
Cofibration and neighborhood deformation retract
A map f: A → X of topological spaces is a 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, then the image of a cofibration f is closed inX. 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 ifA is a neighborhood deformation retract of X, meaning that there is a continuous map with and a homotopy such that for all for all and and if. For example, the inclusion of a subcomplex in a CW complex is a cofibration.
Properties
One basic property of a retract A of X is that every continuous map has at least one extension namely.
Deformation retraction is a particular case of homotopy equivalence. In fact, two spaces are homotopy equivalent if and only if they are both homeomorphic to deformation retracts of a single larger space.
Any topological space that deformation retracts to a point is contractible and vice versa. However, there exist contractible spaces that do not strongly deformation retract to a point.
No-retraction theorem
The boundary of the n-dimensional ball, that is, the -sphere, is not a retract of the ball.
Absolute neighborhood retract (ANR)
A closed subset of a topological space is called a neighborhood retract of if is a retract of some open subset of that contains. Let be a class of topological spaces, closed under homeomorphisms and passage to closed subsets. Following Borsuk, a space ' is called an absolute retract for the class, written if ' is in and whenever ' is a closed subset of a space in, is a retract of. A space is an absolute neighborhood retract for the class, written if is in and whenever is a closed subset of a space in, is a neighborhood retract of. Various classes such as normal spaces have been considered in this definition, but the class 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 convexmetrizable topological vector space is an AR; more generally, every nonempty convex subset of such a vector space is an AR. For example, any normed vector space is an AR. More concretely, Euclidean space the unit cube and the Hilbert cube are ARs. ANRs form a remarkable class of "well-behaved" topological spaces. Among their properties are:
Every open subset of an ANR is an ANR.
By Hanner, a metrizable space that has an open cover by ANRs is an ANR. It follows that every topological manifold is an ANR. For example, the sphere
' is an ANR but not an AR. In infinite dimensions, Hanner's theorem implies that every Hilbert cube manifold as well as the Hilbert manifolds and Banach manifolds are ANRs.
Every locally finite CW complex is an ANR. An arbitrary CW complex need not be metrizable, but every CW complex has the homotopy type of an ANR.
Counterexamples: Borsuk found a compact subset of that is an ANR but not strictly locally contractible. Borsuk also found a compact subset of the Hilbert cube that is locally contractible but not an ANR.
Every ANR has the homotopy type of a CW complex, by Whitehead and Milnor. Moreover, a locally compact ANR has the homotopy type of a locally finite CW complex; and, by West, a compact ANR has the homotopy type of a finite CW complex. In this sense, ANRs avoid all the homotopy-theoretic pathologies of arbitrary topological spaces. For example, the Whitehead theorem holds for ANRs: a map of ANRs that induces an isomorphism on homotopy groups is a homotopy equivalence. Since ANRs include topological manifolds, Hilbert cube manifolds, Banach manifolds, and so on, these results apply to a large class of spaces.
Many mapping spaces are ANRs. In particular, let Y be an ANR with a closed subspace A that is an ANR, and let X be any compact metrizable space with a closed subspace B. Then the space of maps of pairs is an ANR. It follows, for example, that the loop space of any CW complex has the homotopy type of a CW complex.
By Cauty, a metrizable space is an ANR if and only if every open subset of has the homotopy type of a CW complex.
By Cauty, there is a metric linear space that is not an AR. One can take to be separable and an F-space. Since is contractible and not an AR, it is also not an ANR. By Cauty's theorem above, has an open subset that is not homotopy equivalent to a CW complex. Thus there is a metrizable space that is strictly locally contractible but is not homotopy equivalent to a CW complex. It is not known whether a compact metrizable space that is strictly locally contractible must be an ANR.
