Difference between revisions of "Deformation retract"

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{{redirect|Retract|other meanings including concepts in group theory and category theory|Retraction (disambiguation)}}
 
{{redirect|Retract|other meanings including concepts in group theory and category theory|Retraction (disambiguation)}}
In [[topology]], a branch of mathematics, a '''retraction''',<ref>{{cite journal|title=Sur les rétractes|author=K. Borsuk|journal=Fund. Math.|volume=17|year=1931|pages=2–20}}</ref> is a continuous mapping from the entire [[space (mathematics)|space]] into a [[Subspace topology|subspace]] which preserves the position of all points in that subspace. A '''deformation retraction''' is a [[function (mathematics)|map]] which captures the idea of ''[[continuous function|continuously]] shrinking'' a space into a subspace.
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In [[topology]], a branch of mathematics, a '''retraction'''<ref>{{cite journal|title=Sur les rétractes|author=K. Borsuk|journal=Fund. Math.|volume=17|year=1931|pages=2–20}}</ref> is a continuous mapping from the entire [[space (mathematics)|space]] into a [[Subspace topology|subspace]] which preserves the position of all points in that subspace. A '''deformation retraction''' is a [[function (mathematics)|map]] which captures the idea of ''[[continuous function|continuously]] shrinking'' a space into a subspace.
  
 
== Definitions ==
 
== Definitions ==
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A closed subspace ''A'' is a '''neighborhood deformation retract''' of ''X'' if there exists a continuous map <math>u:X \rightarrow I</math> (where <math>I=[0,1]</math>) such that <math>A = u^{-1} (0)</math> and a homotopy
 
A closed subspace ''A'' is a '''neighborhood deformation retract''' of ''X'' if there exists a continuous map <math>u:X \rightarrow I</math> (where <math>I=[0,1]</math>) such that <math>A = u^{-1} (0)</math> and a homotopy
 
<math>H:X\times I\rightarrow X</math> such that <math>H(x,0) = x</math> for all <math>x \in X</math>, <math>H(a,t) = a</math> for all
 
<math>H:X\times I\rightarrow X</math> such that <math>H(x,0) = x</math> for all <math>x \in X</math>, <math>H(a,t) = a</math> for all
<math>(a,t) \in A\times I</math>, and <math>h(x,1) \in A</math> for all <math>x \in u^{-1} [ 0 , 1)</math>.<ref name='steenrod'>{{cite journal | journal= Michigan Math. J. | last1=Steenrod | first1=N. E. | title=A convenient category of topological spaces | volume=14 | issue=2 | year=1967 | pages=133–152}}</ref>
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<math>(a,t) \in A\times I</math>, and <math>h(x,1) \in A</math> for all <math>x \in u^{-1} [ 0 , 1)</math>.<ref name='steenrod'>{{cite journal | journal= Michigan Math. J. | last1=Steenrod | first1=N. E. | title=A convenient category of topological spaces | volume=14 | issue=2 | year=1967 | pages=133–152 | doi=10.1307/mmj/1028999711}}</ref>
  
 
==Properties==
 
==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.  
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* The main obvious property of a retract ''A'' of ''X'' is that ''any'' continuous map <math>f : A \rightarrow Y</math> has at least one ''extension'' <math>g : X \rightarrow Y</math>, namely, <math>g=f\circ r\,</math>.
  
Any topological space which deformation retracts to a point is [[contractible space|contractible]] and vice versa. However, there exist contractible spaces which do not strongly deformation retract to a point.<ref name='hatcher'>{{Citation | last1=Hatcher | first1=Allen | title=Algebraic topology | url=http://www.math.cornell.edu/~hatcher/AT/ATpage.html | publisher=[[Cambridge University Press]] | isbn=978-0-521-79540-1 | year=2002}}</ref>
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* 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.
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* Any topological space which deformation retracts to a point is [[contractible space|contractible]] and vice versa. However, there exist contractible spaces which do not strongly deformation retract to a point.<ref name='hatcher'>{{Citation | last1=Hatcher | first1=Allen | title=Algebraic topology | url=http://www.math.cornell.edu/~hatcher/AT/ATpage.html | publisher=[[Cambridge University Press]] | isbn=978-0-521-79540-1 | year=2002}}</ref>
  
 
==Notes==
 
==Notes==
 
{{Reflist}}
 
{{Reflist}}
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== References ==
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*J.P. May, A concise course in algebraic topology
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*[[James Munkres|Munkres, James]]; ''Topology'', Prentice Hall; 2nd edition (December 28, 1999). ISBN 0-13-181629-2.
  
 
==External links==
 
==External links==

Latest revision as of 03:27, 30 May 2014

{{#invoke:Hatnote|hatnote}}Template:Main other 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

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, 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 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 is a retraction, then the composition 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.

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

If there exists an open set U such that

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

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 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

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

Neighborhood deformation retract

A closed subspace A is a neighborhood deformation retract of X if there exists a continuous map (where ) such that and a homotopy such that for all , for all , and for all .[2]

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.[3]

Notes

  1. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
  2. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
  3. {{#invoke:citation/CS1|citation |CitationClass=citation }}

References

  • J.P. May, A concise course in algebraic topology
  • Munkres, James; Topology, Prentice Hall; 2nd edition (December 28, 1999). ISBN 0-13-181629-2.

External links