§8 Fundamental Groups

Definition 8.1 Let $X\subset\mathbb{R}^{n}$ be a subset, and fix $x_{0}\in X$ . A loop in $X$ based at $x_{0}$ is a continuous function

[0,1]\xrightarrow{\gamma}\mathbb{R}^{n}

satisfying the following conditions:

For every $t\in[0,1]$ , we have $\gamma(t)\in X$ .
$\gamma(0)=\gamma(1)=x_{0}$ .

Example 8.1 Let

X=\mathbb{R}^{n}\setminus\{0\}, \qquad x_{0}=(1,0).

Definition 8.2 Given two curves $\gamma_{1}$ and $\gamma_{2}$ , we say that $\gamma_{1}$ and $\gamma_{2}$ are homotopic if $\gamma_{1}$ can be continuously deformed into $\gamma_{2}$ without changing $\gamma(0)$ or $\gamma(1)$ .

That is, there exists a continuous map

\begin{aligned} \Gamma:[0,1]\times[a,b]&\to X\\ (t,s)&\mapsto\Gamma(t,s) \end{aligned}\tag*{}

such that

$\Gamma(t,a)=\gamma_{1}(t)$ .
$\Gamma(t,b)=\gamma_{2}(t)$ .
For every $s$ ,

\Gamma(0,s)=\Gamma(1,s)=x_{0}.

Here $[a,b]$ is some interval.

You may think of this as a “path animation” lasting $b-a$ seconds.

Lemma 8.1 If

\gamma_{1}\sim\gamma_{2}

if and only if $\gamma_{1}$ and $\gamma_{2}$ are homotopic, then $\sim$ is an equivalence relation.

Informally:

Every path can be deformed back to itself by “doing no deformation”.
If $\gamma_{1}$ can be deformed into $\gamma_{2}$ , then $\gamma_{2}$ can be deformed back into $\gamma_{1}$ by reversing the deformation.
If $\gamma_{1}$ can be deformed into $\gamma_{2}$ and $\gamma_{2}$ can be deformed into $\gamma_{3}$ , then $\gamma_{1}$ can be deformed into $\gamma_{3}$ by performing the two deformations consecutively.

Definition 8.3 Define

\pi_{1}(X,x_{0}) = \{\text{loops based at }x_{0}\}/\sim.\tag*{}

This is called the fundamental group of $X$ based at $x_{0}$ .

Remark If $X$ is connected, then for any two base points,

\pi_{1}(X,x_{0}) \cong \pi_{1}(X,x_{0}^{\prime}).

Composition in the fundamental group is defined as follows:

\begin{aligned} \pi_{1}(X,x_{0})\times\pi_{1}(X,x_{0}) &\to \pi_{1}(X,x_{0})\\ ([\gamma_{b}],[\gamma_{a}]) &\mapsto [\gamma_{b}][\gamma_{a}]. \end{aligned}\tag*{}

Given two paths $\gamma_{a}$ and $\gamma_{b}$ , consider the path

\begin{aligned} \widetilde{\gamma_{b}\circ\gamma_{a}}:[0,2]&\to X\\ t&\mapsto \begin{cases} \gamma_{a}(t)&\text{if }t\in[0,1]\\ \gamma_{b}(t-1)&\text{if }t\in[1,2]. \end{cases} \end{aligned}\tag*{}

Rescale the interval $[0,2]$ to $[0,1]$ to obtain the path

\begin{aligned} \gamma_{b}\circ\gamma_{a}:[0,1]&\to X\\ t&\mapsto \begin{cases} \gamma_{a}(2t)&\text{if }t\in\left[0,\frac{1}{2}\right]\\ \gamma_{b}(2t-1)&\text{if }t\in\left[\frac{1}{2},1\right]. \end{cases} \end{aligned}\tag*{}

Define

[\gamma_{b}][\gamma_{a}] = [\gamma_{b}\circ\gamma_{a}].

Example 8.2 Consider

X=\mathbb{R}^{2}\setminus\{p,q\},

with $x_{0}$ an arbitrary point.

The two diagrams on the right both represent

[\gamma_{b}\circ\gamma_{a}].

Remark

The constant loop

\gamma(t)=x_{0}

for every $t$ is the identity element.

If $\gamma$ is a loop and $\gamma_{0}$ is the constant path, then $\gamma_{0}\circ\gamma$ is the path that “runs through $\gamma$ quickly and then remains stationary for $\dfrac{1}{2}$ second”.

Let $\Gamma$ be the homotopy that continuously shrinks this $\dfrac{1}{2}$ second of stationary time to $0$ seconds.

The inverse path of $\gamma$ is the loop obtained by “running $\gamma$ backwards”.

$[\text{“}\gamma~\text{backwards”}\circ\gamma]?$

Finally, path composition is associative.

Example 8.3 Notice that $L$ denotes a line.

We do not prove the isomorphism here, but roughly speaking, we can send a path $\gamma$ to its winding number.

“How many times does the path $\gamma$ wind around the hole?”

Example 8.4 If the path $\gamma$ is differentiable, this can be defined by

\dfrac{1}{2\pi\mathrm{i}} \displaystyle\int_{\gamma} \dfrac{\mathrm{d}z}{z}.\tag*{}

Example 8.5

Contents

§8 Fundamental Groups