Topological retraction

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

Retraction/Definition
Claim 1:

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

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

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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
  • $\forall p\in A$ the induced homomorphism on fundamental groups of the retraction, $r_*:\pi_1(X,p)\rightarrow\pi_1(A,p)$ is surjective
• For the inclusion map of a retract of a space the induced homomorphism on the fundamental group is injective
  • $\forall p\in A$ the induced homomorphism on fundamental groups of the inclusion map, $i_A:A\hookrightarrow X$, which is $(i_A)_*:\pi_1(A,p)\rightarrow \pi_1(X,p)$ 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