# Topological retraction

The message provided is:

## Contents

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

- We see:

- By if the composition of two functions is a bijection then the initial map is injective and the latter map is surjective

**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?

# OLD PAGE

**Stub grade: A***

## 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 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

## References

- ↑
^{1.0}^{1.1}Introduction to Topological Manifolds - John M. Lee