# 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 \(t\in[0,1]\).

*loop*(it starts and ends at the same point \((1,0)\).

*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:

*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.*

*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.

## 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*\(t\mapsto 0\).

*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.