Difference between revisions of "Topological retraction"
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{{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]]}} | ||
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'''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}} | ||
Revision as of 08:04, 13 December 2016
Stub grade: A*
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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 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
