1.02 Paths, loops, and homotopies

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


(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.


(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.

If \(x\in X\) 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

  1. Is the homotopy \(\gamma_R\) from the previous video a based homotopy or a free homotopy?


Comments, corrections and contributions are very welcome; please drop me an email at j.d.evans at lancaster.ac.uk if you have something to share.

CC-BY-SA 4.0 Jonny Evans.