Topological retraction

Grade: A*

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Messy as ... - it's just awful

Proof

Note that if $r\circ i_A\eq \text{Id}_A$ then $r_*\circ(i_A)_*\eq (\text{Id}_A)_*$

• So $r_*\circ(i_A)_*$ 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:
      1. $r_*:\pi_1(X,a)\rightarrow\pi_1(A,a)$ is surjective
      2. $(i_A)_*:\pi_1(A,a)\rightarrow\pi_1(X,a)$ is injective

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

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

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

If there is such a retraction, we say that: $A$ is a retract of $X$.
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 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
  • $\forall p\in A$ 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