# Math4202 Topology II (Lecture 19)

## Exam announcement

Cover from first lecture to the fundamental group of circle.

## Algebraic Topology

### Retraction and fixed point

#### Definition of retraction

If $A\subseteq X$, a retraction of $X$ onto $A$ is a continuous map $r:X\to A$ such that $r|_A$ is the identity map of $A$.

When such a retraction $r$ exists, $A$ is called a retract of $X$.

<details>
<summary>Example</summary>

Identity map is a retraction of $X$ onto $X$.

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$X=\mathbb{R}^2$, $A=\{0\}$, the constant map that maps all points to $(0,0)$ is a retraction of $X$ onto $A$.

This can be generalized to any topological space, take $A$ as any one point set in $X$.

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Let $X=\mathbb{R}^2$, $A=\mathbb{R}$, the projection map that maps all points to the first coordinate is a retraction of $X$ onto $A$.

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> Can we retract $\mathbb{R}^2$ to a circle?

Let $\mathbb{R}^2\to S^1$

This can be done in punctured plane. $\mathbb{R}^2\setminus\{0\}\to S^1$. by $\vec{x}\mapsto \vec{x}/\|x\|$.

But

</details>

#### Lemma for retraction

If $A$ is a retract of $X$, the homomorphism of fundamental groups induced by the inclusion map $j:A\to X$, with induced $j_*:\pi_1(A,x_0)\to \pi_1(X,x_0)$ is injective.

<details>
<summary>Proof</summary>

Let $r:X\to A$ be a retraction. Consider $j:A\to X, r:X\to A$. Then $r\circ j(a)=r(a)=a$. Therefore $r\circ j=Id_A$.

Then $r_*\circ j_*=Id_{\pi_1(A,x_0)}$.

$\forall f\in \ker j_*$, $j_*f=0$. $r_*\circ j_*f=Id_{f}=f$, therefore $f=0$.

So $\ker j_*=\{0\}$.

So it is injective.
</details>

Consider the $\mathbb{R}^2\to S^1$ example, if such retraction exists, $j_*:\pi_1(S^1,x_0)\to \pi_1(\mathbb{R}^2,x_0)$ is injective. But the fundamental group of circle is $\mathbb{Z}$ whereas the fundamental group of plane is $1$. That cannot be injective.

#### Corollary for lemma of retraction

There is no retraction from $\mathbb{R}^2$, $B_1(0)\subseteq \mathbb{R}^2$ (unit ball in $\mathbb{R}^2$), to $S^1$.

#### Lemma

Let $h:S^1\to X$ be a continuous map. The following are equivalent:

- $h$ is null-homotopic ($h$ is homotopic to a constant map).
- $h$ extends to a continuous map from $B_1(0)\to X$.
- $h_*$ is the trivial group homomorphism of fundamental groups (Image of $\pi_1(S^1,x_0)\to \pi_1(X,x_0)$ is trivial group, identity).