Math4202 Topology II (Lecture 7)

Algebraic Topology

Classify 2-dimensional topological manifolds (connected) up to homeomorphism/homotopy equivalence.

Use fundamental groups.

We want to show that:

The fundamental group is invariant under the equivalence relation.
develop some methods to compute the groups.
2-dimensional topological spaces with the same fundamental group are equivalent (homeomorphism).

Homotopy of paths

Definition of path

If $f$ and $f'$ are two continuous maps from $X$ to $Y$ , where $X$ and $Y$ are topological spaces. Then we say that $f$ is homotopic to $f'$ if there exists a continuous map $F:X\times [0,1]\to Y$ such that $F(x,0)=f(x)$ and $F(x,1)=f'(x)$ for all $x\in X$ .

The map $F$ is called a homotopy between $f$ and $f'$ .

We use $f\simeq f'$ to mean that $f$ is homotopic to $f'$ .

Definition of homotopic equivalence map

Let $f:X\to Y$ and $g:Y\to X$ be two continuous maps. If $f\circ g:Y\to Y$ and $g\circ f:X\to X$ are homotopic to the identity maps $\operatorname{id}_Y$ and $\operatorname{id}_X$ , then $f$ and $g$ are homotopic equivalence maps. And the two spaces $X$ and $Y$ are homotopy equivalent.

Note

This condition is weaker than homeomorphism. (In homeomorphism, let $g=f^{-1}$ , we require $g\circ f=\operatorname{id}_X$ and $f\circ g=\operatorname{id}_Y$ .)

Example of homotopy equivalence maps

Let $X=\{a\}$ and $Y=[0,1]$ with standard topology.

Consider $f:X\to Y$ by $f(a)=0$ and $g:Y\to X$ by $g(y)=a$ , where $y\in [0,1]$ .

$g\circ f=\operatorname{id}_X$ and $f\circ g=[0,1]\mapsto 0$ .

$g\circ f\simeq \operatorname{id}_X$

and $f\circ g\simeq \operatorname{id}_Y$ .

Consider $F:X\times [0,1]\to Y$ by $F(a,0)=0$ and $F(a,t)=(1-t)y$ . $F$ is continuous and homotopy between $f\circ g$ and $\operatorname{id}_Y$ .

This gives example of homotopy but not homeomorphism.

Definition of null homology

If $f:X\to Y$ is homotopy to a constant map. $f$ is called null homotopy.

Definition of path homotopy

Let $f,f':I\to X$ be a continuous maps from an interval $I=[0,1]$ to a topological space $X$ .

Two pathes $f$ and $f'$ are path homotopic if

there exists a continuous map $F:I\times [0,1]\to X$ such that $F(i,0)=f(i)$ and $F(i,1)=f'(i)$ for all $i\in I$ .
$F(s,0)=f(0)$ and $F(s,1)=f(1)$ , $\forall s\in I$ .