Deformation retract

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

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

Definitions[edit]

Retract[edit]

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). If X is Hausdorff, then A must be closed.

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 space X is known as an absolute retract if for every normal space Y that contains X as a closed subspace, X is a retract of Y. The unit cube In as well as the Hilbert cube Iω are absolute retracts.

Neighborhood retract[edit]

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. The n-sphere Sn is an absolute neighborhood retract.

Deformation retract and strong deformation retract[edit]

A continuous map

F: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, \; F(x,1) \in A ,\quad \mbox{and} \quad F(a,1) = a \mbox{ for every } 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 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: 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

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 Allen 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

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

Neighborhood deformation retract[edit]

A closed subspace A is a neighborhood deformation retract of X if there exists a continuous map u:X \rightarrow I (where I=[0,1]) such that A = u^{-1} (0) and a homotopy H:X\times I\rightarrow X such that H(x,0) = x for all x \in X, H(a,t) = a for all (a,t) \in A\times I, and h(x,1) \in A for all x \in u^{-1} [ 0 , 1).[2]

Properties[edit]

  • The main obvious property of a retract A of X is that any continuous map f : A \rightarrow Y has at least one extension g : X \rightarrow Y, namely, g=f\circ r\,.
  • 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.[3]

Notes[edit]

  1. ^ K. Borsuk (1931). "Sur les rétractes". Fund. Math. 17: 2–20. 
  2. ^ Steenrod, N. E. (1967). "A convenient category of topological spaces". Michigan Math. J. 14 (2): 133–152. doi:10.1307/mmj/1028999711. 
  3. ^ Hatcher, Allen (2002), Algebraic topology, Cambridge University Press, ISBN 978-0-521-79540-1 

References[edit]

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