Topological retraction

Grade: A*

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Messy as ... - it's just awful

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:
      1. [ilmath]r_*:\pi_1(X,a)\rightarrow\pi_1(A,a)[/ilmath] is surjective
      2. [ilmath](i_A)_*:\pi_1(A,a)\rightarrow\pi_1(X,a)[/ilmath] is injective

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?

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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 [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:

• 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 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]

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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

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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
  • [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