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).

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