# 1.02 Paths, loops, and homotopies

## Video

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

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**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.

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 \(\gamma_1\).

**Figure 2.** *A picture of a typical based homotopy. The basepoint is
the black dot on the right; both top and bottom edges of the domain*
*map to the basepoint.*

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.