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> | + | 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 | + | 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''' | + | 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 | + | 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.) | ||
− | + | 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 | |
− | + | :<math>F(x,t)=\left((1-t)+{t\over \|x\|}\right) x.</math> | |
− | <math> | + | |
− | <math>(t | + | ===Neighborhood deformation retract=== |
− | + | 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]]. | + | 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]] | ||
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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: 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
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
External links
- This article incorporates material from Neighborhood retract on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.