Difference between revisions of "Deformation retraction/Definition"
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==Definition== | ==Definition== | ||
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| − | A [[subspace (topology)|subspace]], {{M|A}}, of a [[topological space]] {{Top.|X|J}} is called a ''deformation retract'' of {{M|X}}, if there exists a [[retraction]]{{rAITATJJR}}, {{M|r:X\rightarrow A}}, with the additional property: | + | A [[subspace (topology)|subspace]], {{M|A}}, of a [[topological space]] {{Top.|X|J}} is called a ''deformation retract'' of {{M|X}}, if there exists a [[retraction]]{{rAITATJJR}}{{rITTMJML}}, {{M|r:X\rightarrow A}}, with the additional property: |
| − | * {{M|i_A\circ r\simeq\text{Id}_X}}<ref name="AITATJJR"/> | + | * {{M|i_A\circ r\simeq\text{Id}_X}}<ref name="AITATJJR"/><ref name="ITTMJML"/> (That {{M|i_A\circ r}} and {{M|\text{Id}_X}} are [[homotopic maps]]) |
| + | *: Here {{M|i_A:A\hookrightarrow X}} is the [[inclusion map]] and {{M|\text{Id}_X}} the [[identity map]] of {{M|X}}. | ||
| + | Recall that a [[retraction]], {{M|r:X\rightarrow A}} is simply a continuous map where {{M|1=r\vert_A=\text{Id}_A}} (the [[restriction]] of {{M|r}} to {{M|A}}). This is equivalent to the requirement: {{M|1=r\circ i_A=\text{Id}_A}}. | ||
{{#if:{{{hideCaution|}}}|| | {{#if:{{{hideCaution|}}}|| | ||
: {{Caution|Be sure to see the '''''[[Deformation retraction#Warnings on terminology|warnings on terminology]]'''''}}<!-- | : {{Caution|Be sure to see the '''''[[Deformation retraction#Warnings on terminology|warnings on terminology]]'''''}}<!-- | ||
-->}}<noinclude> | -->}}<noinclude> | ||
==References== | ==References== | ||
| + | {{Todo|Mention something about how we must have a [[homotopy equivalence]] as a result. If {{M|1=r\circ i_A=\text{Id}_A}} then {{M|r\circ i_A}} and {{M|\text{Id}_X}} are trivially homotopic. As {{M|i_A\circ r\simeq\text{Id}_A}} we have the definition of a [[homotopy equivalence]]}} | ||
<references/> | <references/> | ||
{{Definition|Topology|Homotopy Theory|Algebraic Topology}} | {{Definition|Topology|Homotopy Theory|Algebraic Topology}} | ||
</noinclude> | </noinclude> | ||
Revision as of 20:09, 12 May 2016
Definition
A subspace, [ilmath]A[/ilmath], of a topological space [ilmath](X,\mathcal{ J })[/ilmath] is called a deformation retract of [ilmath]X[/ilmath], if there exists a retraction[1][2], [ilmath]r:X\rightarrow A[/ilmath], with the additional property:
- [ilmath]i_A\circ r\simeq\text{Id}_X[/ilmath][1][2] (That [ilmath]i_A\circ r[/ilmath] and [ilmath]\text{Id}_X[/ilmath] are homotopic maps)
- Here [ilmath]i_A:A\hookrightarrow X[/ilmath] is the inclusion map and [ilmath]\text{Id}_X[/ilmath] the identity map of [ilmath]X[/ilmath].
Recall that a retraction, [ilmath]r:X\rightarrow A[/ilmath] is simply a continuous map where [ilmath]r\vert_A=\text{Id}_A[/ilmath] (the restriction of [ilmath]r[/ilmath] to [ilmath]A[/ilmath]). This is equivalent to the requirement: [ilmath]r\circ i_A=\text{Id}_A[/ilmath].
- Caution:Be sure to see the warnings on terminology
References
TODO: Mention something about how we must have a homotopy equivalence as a result. If [ilmath]r\circ i_A=\text{Id}_A[/ilmath] then [ilmath]r\circ i_A[/ilmath] and [ilmath]\text{Id}_X[/ilmath] are trivially homotopic. As [ilmath]i_A\circ r\simeq\text{Id}_A[/ilmath] we have the definition of a homotopy equivalence
