AskDefine | Define retract

Dictionary Definition

retract

Verb

1 formally reject or disavow a formerly held belief, usually under pressure; "He retracted his earlier statements about his religion"; "She abjured her beliefs" [syn: abjure, recant, forswear, resile]
2 pull away from a source of disgust or fear [syn: shrink back]
3 use a surgical instrument to hold open (the edges of a wound or an organ) [syn: pull back, draw back]
4 pull inward or towards a center; "The pilot drew in the landing gear"; "The cat retracted his claws" [syn: draw in]

User Contributed Dictionary

English

Etymology

Pronunciation

(US) IPA: /rəˈtrækt/

Verb

  1. To pull back inside (for example, an airplane retracting its wheels while flying).
  2. To take back or withdraw something one has said.
    I retract all the accusations I made about the senator and sincerely hope he won't sue me.

Extensive Definition

In topology, a retraction, as the name suggests, "retracts" an entire space into a subspace. A deformation retraction is a map which captures the idea of continuously shrinking a space into a subspace.

Definitions

Retract

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.
A space X is known as an absolute retract (or AR) if for every normal space Y that embeds X as a closed subset, X is a retract of Y.

Neighborhood retract

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.

Deformation retract and Strong deformation retract

A continuous map
d:X \times [0, 1] \to X
is a deformation retraction if, for every x in X and a in A,
d(x,0) = x, \; d(x,1) \in A ,\quad \mbox \quad d(a,1) = a.
In other words, a deformation retraction is a homotopy between a retract and the identity map on X. The subspace A is called a deformation retract of X. A deformation retract 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.
Note: An equivalent definition of deformation retraction is the following. A continuous map r: X → 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
d(a,t) = a\,
for all t in [0, 1], d is called a strong deformation retraction. In other words, a strong deformation retraction leaves points in A fixed throughout the homotopy. (Some authors take this as the definition of deformation retraction.)

Neighborhood deformation retract

A pair (X, A) of spaces in U is an NDR-pair if there exists a map u:X \rightarrow I such that A = u^ (0) and a homotopy h:I \times X \rightarrow X such that h(0, x) = x for all x \in X, h(t, a) = a for all (t, a) \in I \times A, and h(1, x) \in A for all x \in u^ [ 0 , 1). The pair (h, u) is said to be a representation of (X, A) as an NDR-pair.

Properties

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.

References

retract in German: Retraktion
retract in Finnish: Deformaatioretrakti
retract in Italian: Retrazione
retract in Polish: Retrakt deformacyjny
retract in Russian: Деформационный ретракт

Synonyms, Antonyms and Related Words

abandon, abjure, abolish, abrogate, annul, assert the contrary, back, back down, back out, backwater, belie, cancel, cede, climb down, contest, contradict, contravene, controvert, counter, countermand, counterorder, crawfish out, cringe, cross, deny, disaffirm, disallow, disannul, disavow, disclaim, disgorge, disown, dispense with, dispose of, disprove, dispute, do away with, do without, draw back, draw in, drop, duck, dump, eat crow, eat humble pie, fall back, flinch, forgo, forswear, gainsay, get along without, get rid of, give away, give up, have done with, impugn, invalidate, join issue upon, kiss good-bye, make a sacrifice, make void, not accept, not admit, nullify, oppose, override, overrule, palinode, part with, pull back, pull in, pull out, quitclaim, recall, recant, refuse to admit, refute, relinquish, render up, renege, renounce, repeal, repudiate, rescind, resign, retreat, retrocede, retrograde, reverse, revoke, sacrifice, set aside, shrink, shy, spare, surrender, suspend, swallow, swear off, take, take back, take issue with, throw up, unsay, vacate, void, waive, wince, withdraw, write off, yield
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