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> as the name suggests, "retracts" an entire [[space (mathematics)|space]] into a [[Subspace topology|subspace]]. A '''deformation retraction''' is a [[function (mathematics)|map]] which captures the idea of ''[[continuous function|continuously]] shrinking'' a space into a subspace.
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 ==
=== Retract ===
=== Retract ===
Let ''X'' be a [[topological space]] and ''A'' a subspace of ''X''. Then a continuous map  
Let ''X'' be a [[topological space]] and ''A'' a [[subspace (topology)|subspace]] of ''X''. Then a continuous map  


:<math>r:X \to A</math>
:<math>r:X \to A</math>


is a ''retraction'' if the [[function (mathematics)#Restrictions and extensions|restriction]] of ''r'' to ''A'' is the [[Identity function|identity map]] on ''A''; that is, ''r''(''a'') = ''a'' for all ''a'' in ''A''. Equivalently, denoting by  
is a '''retraction''' if the [[function (mathematics)#Restrictions and extensions|restriction]] of ''r'' to ''A'' is the [[Identity function|identity map]] on ''A''; that is, ''r''(''a'') = ''a'' for all ''a'' in ''A''. Equivalently, denoting by  


:<math>\iota : A \hookrightarrow X</math>
:<math>\iota : A \hookrightarrow X</math>
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:<math>r \circ \iota = id_A,</math>
:<math>r \circ \iota = id_A,</math>


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.
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 space|Hausdorff]], then ''A'' must be closed.


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''. The unit cube ''I<sup>n</sup>'' as well as the [[Hilbert cube]] ''I<sup>ω</sup>'' are absolute retracts.
If <math>r:X \to A</math> is a retraction, then the composition <math>\iota \circ r</math> is an [[idempotent]] continuous map from ''X'' to ''X''. Conversely, given any idempotent continuous map <math>s:X\to X</math>, 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]] ''I<sup>n</sup>'' as well as the [[Hilbert cube]] ''I<sup>ω</sup>'' are absolute retracts.


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


:<math> F(x,0) = x, \; F(x,1) \in A ,\quad \mbox{and} \quad F(a,1) = a.</math>
:<math> F(x,0) = x, \; F(x,1) \in A ,\quad \mbox{and} \quad F(a,1) = a \mbox{ for every } a \in A .</math>


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 retract is a special case of [[homotopy equivalence]].
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).
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).
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for all ''t'' in [0, 1], ''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.)
for all ''t'' in [0, 1], ''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.)


==Neighborhood deformation retract==
As an example, the [[n-sphere|''n''-sphere]] ''S<sup>n</sup>'' is a strong deformation retract of '''R'''<sup>''n''+1</sup>\{0}; as strong deformation retraction one can choose the map
A pair <math>(X, A)</math> of spaces in U is an NDR-pair if there exists a map <math>u:X \rightarrow I</math> such that <math>A = u^{-1} (0)</math> and a homotopy
:<math>F(x,t)=\left((1-t)+{t\over \|x\|}\right) x.</math>
<math>h:I \times X \rightarrow X</math> such that <math>h(0, x) = x</math> for all <math>x \in X</math>, <math>h(t, a) = a</math> for all
 
<math>(t, a) \in I \times A</math>, and <math>h(1, x) \in A</math> for all <math>x \in u^{-1} [ 0 , 1)</math>. The pair <math>(h, u)</math> is said to
===Neighborhood deformation retract===
be a representation of <math>(X, A)</math> as an NDR-pair.
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>(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>


==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.  
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 space|contractible]]. Contractibility, however, is a weaker condition, as contractible spaces exist which do not 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>
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==
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[[Category:Topology]]
[[Category:Topology]]
[[de:Retraktion und Koretraktion]]
[[fi:Deformaatioretrakti]]
[[it:Retrazione]]
[[pl:Retrakt deformacyjny]]
[[ru:Деформационный ретракт]]
[[zh:形变收缩]]

Revision as of 20:08, 11 September 2013

{{#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], 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 }}

External links