Difference between revisions of "Topological retraction"
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| + | {{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]]}} | ||
| − | ==[[ | + | ==[[/Definition|Definition]]== |
| − | {{ | + | {{/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:
The message provided is:
Messy as ... - it's just awful
Contents
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]
- So [ilmath]r_*\circ(i_A)_*[/ilmath] 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:
- [ilmath]r_*:\pi_1(X,a)\rightarrow\pi_1(A,a)[/ilmath] is surjective
- [ilmath](i_A)_*:\pi_1(A,a)\rightarrow\pi_1(X,a)[/ilmath] is injective
- We see:
- By if the composition of two functions is a bijection then the initial map is injective and the latter map is surjective
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
- Types of topological retractions - comparing retraction with deformation retraction and strong deformation retraction
Important theorems
- For a retraction the induced homomorphism on the fundamental group is surjective
- [ilmath]\forall p\in A[/ilmath] the induced homomorphism on fundamental groups of the retraction, [ilmath]r_*:\pi_1(X,p)\rightarrow\pi_1(A,p)[/ilmath] is surjective
- For the inclusion map of a retract of a space the induced homomorphism on the fundamental group is injective
- [ilmath]\forall p\in A[/ilmath] the induced homomorphism on fundamental groups of the inclusion map, [ilmath]i_A:A\hookrightarrow X[/ilmath], which is [ilmath](i_A)_*:\pi_1(A,p)\rightarrow \pi_1(X,p)[/ilmath] is injective
Lesser theorems
- A retract of a connected space is connected
- A retract of a compact space is compact
- A retract of a retract of X is a retract of X
- A retract of a simply connected space is simply connected
