Deformation retract

In topology, 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.

Let X be a topological space and A a subspace of X. Then a continuous map

\(r\colon X \to A\)

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

\(\iota\colon A \hookrightarrow X\)

the inclusion, a retraction is a continuous map r such that

\(r \circ \iota = \operatorname{id}_A,\)

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 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 \(\iota \circ r\) 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.

A continuous map

\(F\colon X \times [0, 1] \to X\)

is a deformation retraction of a space X onto a subspace A if, for every x in X and a in A,

\(F(x,0) = x, \quad F(x,1) \in A ,\quad \mbox{and} \quad F(a,1) = a.\)

In other words, a deformation retraction is a homotopy between a retraction (strictly, between its composition with the inclusion) 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 itself called a deformation retraction if it is a retraction and its composition with the inclusion is homotopic to the identity map on X. In this language, a deformation retraction still carries with it a homotopy between the identity map on X and itself, but we refer to the map \(r\) rather than the homotopy as a deformation retraction.

If, in the definition of a deformation retraction, we add the requirement that

\(F(a,t) = a\)

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.)

\(F(x,t)=(1-t)x+t{x\over \|x\|}.\)

Condensed: the full section is in Wikipedia.

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 \(u: X \rightarrow [0, 1]\) with \(A = u^{-1}\!\left(0\right)\) and a homotopy \(H: X \times [0, 1] \rightarrow X\) such that \(H(x,0) = x\) for all \(x \in X,\) \(H(a,t) = a\) for all \(a \in A\) and \(t \in [0, 1],\) and \(H\left(x,1\right) \in A\) if \(u(x) < 1\).

For example, the inclusion of a subcomplex in a CW complex is a cofibration.

• One basic property of a retract A of X (with retraction \(r: X \to A\)) is that every continuous map \(f: A \rightarrow Y\) has at least one extension \(g: X \rightarrow Y,\) namely \(g = f \circ r\).
• If a subspace is a retract of a space, then the inclusion induces an injection between fundamental groups.
• 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.

The boundary of the n-dimensional ball, that is, the (n−1)-sphere, is not a retract of the ball. (See Brouwer fixed-point theorem § A proof using homology or cohomology.)

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 \(\operatorname{AR} \left(\mathcal{C}\right),\) 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 \(\operatorname{ANR} \left(\mathcal{C}\right),\) 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 \(\operatorname {AR} \left({\mathcal {M}}\right)\) and \(\operatorname {ANR} \left({\mathcal {M}}\right)\).

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:

Condensed: the full section is in Wikipedia.