Difference between revisions of "Topological retraction"
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m (Alec moved page Retraction to Topological retraction without leaving a redirect: Retraction is a thing in category theory too) |
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Revision as of 08:04, 13 December 2016
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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
Retraction/Definition
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
