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 $X$ .

Example

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

$X=\mathbb{R}^2$ , $A=\{0\}$ , 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 $X$ .

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 $A$ .

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

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.

Proof

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.

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).