1.02 Paths, loops, and homotopies

Below the video you will find accompanying notes and some pre-class questions.

Previous video: 1.01 Winding numbers and the fundamental theorem of algebra.
Next video: 1.03 Concatenation and the fundamental group.
Index of all lectures.

Notes

(0.00) In this section, we're going to introduce the idea of continuous paths, loops and homotopies of loops. I will assume you're happy with the notions of topological space and continuous maps between topological spaces. If you haven't come across these ideas, I have a series of videos covering these ideas which you can watch at any point during the module. If you like, you can mentally replace the word ``topological space'' with ``metric space'' until you've watched those videos.

Paths

(1.06) Let

$X$ be a topological space. A path in

$X$ is a continuous map

$\gamma\colon[0,1]\to X$ . You can think of this as a continuous parametric curve

$\gamma(t)$ with parameter

$t\in[0,1]$ .

The map

$\gamma(t)=(t,0)$ is a path in the plane

$X=\mathbf{R}^2$ which moves along the

$x$ -axis from

$(0,0)$ to

$(1,0)$ .

The map

$\gamma(t)=(\cos(2\pi t),\sin(2\pi t))$ is another path in the plane which moves around the unit circle. This is a loop (it starts and ends at the same point

$(1,0)$ .

A path

$\gamma\colon[0,1]\to X$ is called a loop if

$\gamma(0)=\gamma(1)$ . It is said to be a loop based at $x$ if

$\gamma(0)=\gamma(1)=x$ .

Free homotopy of loops

(4.06) Roughly speaking, a free homotopy of loops is a continuous one-parameter family of loops $\gamma_s(t)$ interpolating between two loops $\gamma_0(t)$ and $\gamma_1(t)$ . More precisely:

A free homotopy of loops is a continuous map

$H\colon [0,1]\times[0,1]\to X$ such that

$\gamma_s(t):=H(s,t)$ is a loop for each fixed

$s\in[0,1]$ , that is

$H(s,0)=H(s,1)$ for all

$s\in[0,1]$ .

Figure 1. In this figure, we see the domain and target of a free homotopy $H\colon[0,1]\times[0,1]\to X$ . The loops $\gamma_s(t)=H(s,t)$ are indicated in red for $s=0,\frac{1}{4},\frac{1}{2},\frac{3}{4},1$ . The path traced by basepoints is drawn in thick black; as a visual aid, the path traced by the midpoints of the loops is also drawn in dotted black.

We will often write

$\gamma_s$ instead of

$H$ to emphasise the fact that

$H$ is a family of loops.

Continuity of

$H$ as a map of two variables is what gives us a continuous family of continuous loops.

Based homotopy of loops

(7.53) We will need a slightly more restricted notion of homotopy.

$x\in X$ is a basepoint, a homotopy based at

$x$ is a homotopy

$H\colon[0,1]\times[0,1]\to X$ where

$H(s,0)=H(s,1)=x$ for all

$s\in [0,1]$ . In other words, all the loops

$\gamma_s(t)=H(s,t)$ pass through the basepoint

$x$ at

$t=0,1$ . If we have a based homotopy

$\gamma_s$ we will write

$\gamma_0\simeq\gamma_1$ and say that

$\gamma_0$ is homotopic to

$\gamma_1$ .

By focusing on based loops (and based homotopy) we will be able to concatenate loops (because they all start and end at the same point) and this will give us a group structure on (homotopy classes of) loops.

Loops in $\mathbf{R}^n$ are contractible

(11.04) If

$\gamma\colon[0,1]\to\mathbf{R}^n$ is a loop based at the origin

$0$ then

$\gamma$ is based-homotopic to the constant loop

$t\mapsto 0$ .

The map

$H(s,t)=(1-s)\gamma(t)$ is a continuous map such that

$H(s,0)=0=H(s,1)$ , so it is a based homotopy. At

$s=0$ , we get

$\gamma_0(t)=H(0,t)=\gamma(t)$ . At

$s=1$ , we get

$\gamma_1(t)=H(1,t)=0$ , so this is a based homotopy from

$\gamma$ to the constant loop.

A homotopy from a loop

$\gamma$ to a constant loop is called a nullhomotopy (and we say that

$\gamma$ is contractible if it is nullhomotopic).

Pre-class questions

Is the homotopy $\gamma_R$ from the previous video a based homotopy or a free homotopy?

Navigation

Previous video: 1.01 Winding numbers and the fundamental theorem of algebra.
Next video: 1.03 Concatenation and the fundamental group.
Index of all lectures.

1.02 Paths, loops, and homotopies

Notes

Paths

Free homotopy of loops

Based homotopy of loops

Loops in Rn\mathbf{R}^n are contractible

Pre-class questions

Navigation

Loops in $\mathbf{R}^n$ are contractible