Examples

Contractibility of Rn

Contractibility of ${\mathbb{R}}^n$

We show that ${\mathbb{R}}^n$ is contractible.

Let $p=0\in {\mathbb{R}}^n$ be the origin. Consider the maps $f$ and $g$:

Then $f\circ g:{\mathbb{R}}^n\to p$ is the constant map sending everything in ${\mathbb{R}}^n$ to the origin.

Define $F(x,t)=tx$. Then $F(x,0)\equiv 0$, which agrees with $f\circ g$, while $F(x,1)=x={\mathrm{Id}}_{{\mathbb{R}}^n}$. Therefore $f\circ g$ is homotopic to ${\mathrm{Id}}_{{\mathbb{R}}^n}$.

Now consider that $g\circ f={\mathrm{Id}}_p$ already. No additional homotopy is needed.

Cylinder is homotopic to circle

Cylinder is homotopic to circle

A cylinder is given by $X={\mathbb{S}}^1\times [-1,1]$. Consider the ‘collapse’ map $f:X\to {\mathbb{S}}^1$ given by projecting to the circle along straight lines, and the map $g:{\mathbb{S}}^1\hookrightarrow X$ given by inclusion:

The map $f\circ g$ is equal to the collapse map, and the map $g\circ f$ is the identity on ${\mathbb{S}}^1$.

Now construct a homotopy from the collapse map to ${\mathrm{Id}}_X$:

$$\begin{array}{rl}F:X\times [0,1]& ⟶X\\ & \\ F:({\mathbb{S}}^1\times [-1,1])\times [0,1]& ⟶{\mathbb{S}}^1\times [-1,1]\\ & \\ ((x,h),t)& ⟼(x,th)\end{array}$$