For other meanings including concepts in group theory and category theory, see Retraction (disambiguation).

In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace which preserves the position of all points in that subspace.[1] A deformation retraction is a mapping which 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

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 (the constant map yields a retraction). If X is Hausdorff, then A must be a closed subset of X.

If r: XA is a retraction, then the composition ι∘r is an idempotent continuous map from X to X. Conversely, given any idempotent continuous map s: XX, 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: 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

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 Sn is a strong deformation retract of Rn+1\{0}; as strong deformation retraction one can choose the map

Cofibration and neighborhood deformation retract

A map f: AX 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.[2] 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: XI (where I = [0,1]) with A = u−1(0) and a homotopy H: X × IX such that H(x,0) = x for all xX, H(a,t) = a for all (a,t) ∈ A × I, and h(x,1) ∈ A if u(x) < 1.[3]

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


No-retraction theorem

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.)

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 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 , written AR(), if X is in and whenever X is a closed subset of a space Y in , X is a retract of Y. A space X is an absolute neighborhood retract for the class , written ANR(), if X is in and whenever X is a closed subset of a space Y in , X is a neighborhood retract of Y.

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 AR() and ANR().[5]

A metrizable space is an AR if and only if it is contractible and an ANR.[6] 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.[7] For example, any normed vector space (complete or not) is an AR. More concretely, Euclidean space Rn, the unit cube In, and the Hilbert cube Iω are ARs.

ANRs form a remarkable class of "well-behaved" topological spaces. Among their properties are:


  1. Borsuk (1931).
  2. Hatcher (2002), Proposition 4H.1.
  3. Puppe (1967), Satz 1.
  4. Hatcher (2002), Exercise 0.6.
  5. Mardešiċ (1999), p. 242.
  6. Hu (1965), Proposition II.7.2.
  7. Hu (1965), Corollary II.14.2 and Theorem II.3.1.
  8. Hu (1965), Theorem III.8.1.
  9. Mardešiċ (1999), p. 245.
  10. Fritsch & Piccinini (1990), Theorem 5.2.1.
  11. Hu (1965), Theorem V.7.1.
  12. Borsuk (1967), section IV.4.
  13. Borsuk (1967), Theorem V.11.1.
  14. Fritsch & Piccinini (1990), Theorem 5.2.1.
  15. West (2004), p. 119.
  16. Hu (1965), Theorem VII.3.1 and Remark VII.2.3.
  17. Cauty (1994), Fund. Math. 144: 11–22.
  18. Cauty (1994), Fund. Math. 146: 85–99.


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