Difference between revisions of "Topological retraction"

From Maths
Jump to: navigation, search
(Created page with "{{Stub page|grade=A*|msg=Demote to grade A once tidied up. Find other sources. Be sure to link to deformation retraction and strong deformation retraction}} ==Retrac...")
 
(This page is crap, added refactor notice, proof of main claim, note to self)
 
(3 intermediate revisions by the same user not shown)
Line 1: Line 1:
 +
{{Refactor notice|grade=A*|msg=Messy as ... - it's just awful}}
 +
==Proof==
 +
Note that if {{M|r\circ i_A\eq \text{Id}_A}} then {{M|r_*\circ(i_A)_*\eq (\text{Id}_A)_*}}
 +
* So {{M|r_*\circ(i_A)_*}} must be a [[bijection]]
 +
** By [[if the composition of two functions is a bijection then the initial map is injective and the latter map is surjective]]
 +
*** We see:
 +
***# {{M|r_*:\pi_1(X,a)\rightarrow\pi_1(A,a)}} is [[surjective]]
 +
***# {{M|(i_A)_*:\pi_1(A,a)\rightarrow\pi_1(X,a)}} is [[injective]]
 +
{{Green highlight|1='''Alec's thought: ''' can we use the [[first group isomorphism theorem]] on {{M|r_*}} to get {{M|\pi_1(A,a)}} from {{M|\pi_1(X,a)}} or something?}}
 +
=OLD PAGE=
 
{{Stub page|grade=A*|msg=Demote to grade A once tidied up. Find other sources. Be sure to link to [[deformation retraction]] and [[strong deformation retraction]]}}
 
{{Stub page|grade=A*|msg=Demote to grade A once tidied up. Find other sources. Be sure to link to [[deformation retraction]] and [[strong deformation retraction]]}}
==[[Retraction/Definition|Definition]]==
+
==[[/Definition|Definition]]==
{{:Retraction/Definition}}<br/>
+
{{/Definition}}<br/>
 
'''Claim 1:'''
 
'''Claim 1:'''
 
* This is equivalent to the condition: {{M|1=r\circ i_A=\text{Id}_A}} where {{M|i_A}} denotes the [[inclusion map (topology)|inclusion map]], {{M|i_A:A\hookrightarrow X}} given by {{M|i_A:a\mapsto x}}
 
* This is equivalent to the condition: {{M|1=r\circ i_A=\text{Id}_A}} where {{M|i_A}} denotes the [[inclusion map (topology)|inclusion map]], {{M|i_A:A\hookrightarrow X}} given by {{M|i_A:a\mapsto x}}
 
{{Todo|In the case of {{M|1=A=\emptyset}} - does it matter? I don't think so, but check there is nothing ''noteworthy'' about it. Also proof of claims}}
 
{{Todo|In the case of {{M|1=A=\emptyset}} - does it matter? I don't think so, but check there is nothing ''noteworthy'' about it. Also proof of claims}}
 
==See also==
 
==See also==
* [[Types of retractions]] - comparing ''retraction'' with [[deformation retraction]] and [[strong deformation retraction]]
+
* [[Types of topological retractions]] - comparing ''retraction'' with [[deformation retraction]] and [[strong deformation retraction]]
 
===Important theorems===
 
===Important theorems===
 
* [[For a retraction the induced homomorphism on the fundamental group is surjective]]
 
* [[For a retraction the induced homomorphism on the fundamental group is surjective]]

Latest revision as of 10:49, 14 December 2016

Grade: A*
This page is currently being refactored (along with many others)
Please note that this does not mean the content is unreliable. It just means the page doesn't conform to the style of the site (usually due to age) or a better way of presenting the information has been discovered.
The message provided is:
Messy as ... - it's just awful

Proof

Note that if [ilmath]r\circ i_A\eq \text{Id}_A[/ilmath] then [ilmath]r_*\circ(i_A)_*\eq (\text{Id}_A)_*[/ilmath]

Alec's thought: can we use the first group isomorphism theorem on [ilmath]r_*[/ilmath] to get [ilmath]\pi_1(A,a)[/ilmath] from [ilmath]\pi_1(X,a)[/ilmath] or something?

OLD PAGE

Stub grade: A*
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Demote to grade A once tidied up. Find other sources. Be sure to link to deformation retraction and strong deformation retraction

Definition

Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space and let [ilmath]A\in\mathcal{P}(X)[/ilmath] be considered a s subspace of [ilmath]X[/ilmath]. A continuous map, [ilmath]r:X\rightarrow A[/ilmath] is called a retraction if[1]:

  • The restriction of [ilmath]r[/ilmath] to [ilmath]A[/ilmath] (the map [ilmath]r\vert_A:A\rightarrow A[/ilmath] given by [ilmath]r\vert_A:a\mapsto r(a)[/ilmath]) is the identity map, [ilmath]\text{Id}_A:A\rightarrow A[/ilmath] given by [ilmath]\text{Id}_A:a\mapsto a[/ilmath]

If there is such a retraction, we say that: [ilmath]A[/ilmath] is a retract[1] of [ilmath]X[/ilmath].
Claim 1:

  • This is equivalent to the condition: [ilmath]r\circ i_A=\text{Id}_A[/ilmath] where [ilmath]i_A[/ilmath] denotes the inclusion map, [ilmath]i_A:A\hookrightarrow X[/ilmath] given by [ilmath]i_A:a\mapsto x[/ilmath]

TODO: In the case of [ilmath]A=\emptyset[/ilmath] - does it matter? I don't think so, but check there is nothing noteworthy about it. Also proof of claims


See also

Important theorems

Lesser theorems

References

  1. 1.0 1.1 Introduction to Topological Manifolds - John M. Lee