4.1 Fundamental groups

4.1.1 Basic ideas

Let us look at figure 4.1. One disc has a hole in it, the other does not. What characterizes the difference between these two discs? We note that any loop in figure $4.1(b)$ can be continuously shrunk to a point. In contrast, the loop $\alpha$ in figure $4.1(a)$ cannot be shrunk to a point due to the existence of a hole in it. Some loops in figure $4.1(a)$ may be shrunk to a point while others cannot. We say a loop $\alpha$ is homotopic to $\beta$ if $\alpha$ can be obtained from $\beta$ by a continuous deformation. For example, any loop in $Y$ is homotopic to a point. It turns out that ‘homotopic to’ is an equivalence relation, the equivalence class of which is called the homotopy class. In figure 4.1, there is only one homotopy class associated with $Y$ . In $X$ , each homotopy class is characterized by $n\in \mathbb{Z}$ , $n$ being the number of times the loop encircles the hole; $n<0$ if it winds clockwise, $n>0$ if counterclockwise, $n=0$ if the loop does not wind round the hole. Moreover, $\mathbb{Z}$ is an additive group and the group operation (addition) has a geometrical meaning; $n+m$ corresponds to going round the hole first $n$ times and then $m$ times. The set of homotopy classes is endowed with a group structure called the fundamental group.

Figure 4.1. A disc with a hole $(a)$ and without a hole $(b)$ . The hole in $(a)$ prevents the loop $\alpha$ from shrinking to a point.

4.1.2 Paths and loops

Definition 4.1

Let $X$ be a topological space and let $I=[0,1]$ . A continuous map $\alpha :I\to X$ is called a path with an initial point $x_0$ and an end point $x_1$ if $\alpha (0)=x_0$ and $\alpha (1)=x_1$ . If $\alpha (0)=\alpha (1)=x_0$ , the path is called a loop with base point $x_0$ (or a loop at $x_0$ ).

For $x\in X$ , a constant path $c_x:I\to X$ is defined by $c_x(s)=x,s\in I$ . A constant path is also a constant loop since $c_x(0)=c_x(1)=x$ . The set of paths or loops in a topological space $X$ may be endowed with an algebraic structure as follows.

Definition 4.2

Let $\alpha ,\beta :I\to X$ be paths such that $\alpha (1)=\beta (0)$ . The product of $\alpha$ and $\beta$ , denoted by $\alpha \ast \beta$ , is a path in $X$ defined by

$$\alpha \ast \beta (s)=\{\begin{array}{ll}\alpha (2s)& 0\le s\le \frac{1}{2}\\ \beta (2s-1)& \frac{1}{2}\le s\le 1\end{array}\text{\hspace{1em}(4.1)}$$

see figure 4.2. Since $\alpha (1)=\beta (0)$ , $\alpha \ast \beta$ is a continuous map from $I$ to $X$ . [Geometrically, $\alpha \ast \beta$ corresponds to traversing the image $\alpha (I)$ , in the first half, then followed by $\beta (I)$ in the remaining half. Note that the velocity is doubled.]

Figure 4.2. The product $\alpha \ast \beta$ of paths $\alpha$ and $\beta$ with a common end point.

Definition 4.3

Let $\alpha :I\to X$ be a path from $x_0$ to $x_1$ . The inverse path ${\alpha }^{-1}$ of $\alpha$ is defined by

$${\alpha }^{-1}(s)\equiv \alpha (1-s)s\in I.\text{\hspace{1em}(4.2)}$$

[The inverse path ${\alpha }^{-1}$ corresponds to traversing the image of $\alpha$ in the opposite direction from $x_1$ to $x_0$ .]

Since a loop is a special path for which the initial point and end point agree, the product of loops and the inverse of a loop are defined in exactly the same way.

It seems that a constant map $c_x$ is the unit element. However, it is not: $\alpha \ast {\alpha }^{-1}$ is not equal to $c_x!$ . We need a concept of homotopy to define a group operation in the space of loops.

4.1.3 Homotopy

The algebraic structure of loops introduced earlier is not so useful as it is. For example, the constant path is not exactly the unit element. We want to classify the paths and loops according to a neat equivalence relation so that the equivalence classes admit a group structure. It turns out that if we identify paths or loops that can be deformed continuously one into another, the equivalence classes form a group. Since we are primarily interested in loops, most definitions and theorems are given for loops. However, it should be kept in mind that many statements are also applied to paths with proper modifications.

Definition 4.4

Let $\alpha ,\beta :I\to X$ be loops at $x_0$ . They are said to be homotopic, written as $\alpha \sim \beta$ , if there exists a continuous map $F:I\times I\to X$ such that

$$F(s,0)=\alpha (s),F(s,1)=\beta (s)\forall s\in I\text{\hspace{1em}(4.3)}$$

$$F(0,t)=F(1,t)=x_0\forall t\in I.$$

The connecting map $F$ is called a homotopy between $\alpha$ and $\beta$ .

It is helpful to represent a homotopy as figure 4.3(a). The vertical edges of the square $I\times I$ are mapped to $x_0$ . The lower edge is $\alpha (s)$ while the upper edge is $\beta (s)$ . In the space $X$ , the image is continuously deformed as in figure 4.3(b).

(a)

(b)

Figure 4.3. (a) The square represents a homotopy $F$ interpolating the loops $\alpha$ and $\beta$ . (b) The image of $\alpha$ is continuously deformed to the image of $\beta$ in real space $X$ .

Figure 4.4. A homotopy $H$ between $\alpha$ and $\gamma$ via $\beta$ .

Proposition 4.1

The relation $\alpha \sim \beta$ is an equivalence relation.

Proof

Reflectivity: $\alpha \sim \alpha$ . The homotopy may be given by $F(s,t)=\alpha (s)$ for any $t\in I$ .

Symmetry: Let $\alpha \sim \beta$ with the homotopy $F(s,t)$ such that $F(s,0)=\alpha (s)$ , $F(s,1)=\beta (s)$ . Then $\beta \sim \alpha$ , where the homotopy is given by $F(s,1-t)$ .

Transitivity: Let $\alpha \sim \beta$ and $\beta \sim \gamma$ . Then $\alpha \sim \gamma$ . If $F(s,t)$ is a homotopy between $\alpha$ and $\beta$ and $G(s,t)$ is a homotopy between $\beta$ and $\gamma$ , a homotopy between $\alpha$ and $\gamma$ may be (figure 4.4)

$$H(s,t)=\{\begin{array}{ll}F(s,2t)& 0\le t\le \frac{1}{2}\\ G(s,2t-1)& \frac{1}{2}\le t\le 1.\end{array}$$

4.1.4 Fundamental groups

The equivalence class of loops is denoted by $[\alpha ]$ and is called the homotopy class of $\alpha$ . The product between loops naturally defines the product in the set of homotopy classes of loops.

Definition 4.5

Let $X$ be a topological space. The set of homotopy classes of loops at $x_0\in X$ is denoted by ${\pi }_1(X,x_0)$ and is called the fundamental group (or the first homotopy group) of $X$ at $x_0$ . The product of homotopy classes $[\alpha ]$ and $[\beta ]$ is defined by

$$[\alpha ]\ast [\beta ]=[\alpha \ast \beta ].\text{\hspace{1em}(4.4)}$$

Lemma 4.1

The product of homotopy classes is independent of the representative, that is, if $\alpha \sim {\alpha }^{\prime }$ and $\beta \sim {\beta }^{\prime }$ , then $\alpha \ast \beta \sim {\alpha }^{\prime }\ast {\beta }^{\prime }$ .

Proof

Let $F(s,t)$ be a homotopy between $\alpha$ and ${\alpha }^{\prime }$ and $G(s,t)$ be a homotopy between $\beta$ and ${\beta }^{\prime }$ . Then

$$H(s,t)=\{\begin{array}{ll}F(2s,t)& 0\le s\le \frac{1}{2}\\ G(2s-1,t)& \frac{1}{2}\le s\le 1\end{array}$$

is a homotopy between $\alpha \ast \beta$ and ${\alpha }^{\prime }\ast {\beta }^{\prime }$ , hence $\alpha \ast \beta \sim {\alpha }^{\prime }\ast {\beta }^{\prime }$ and $[\alpha ]\ast [\beta ]$ is well defined.

Theorem 4.1

The fundamental group is a group. Namely, if $\alpha ,\beta ,\dots$ are loops at $x\in X$ , the following group properties are satisfied:

(1) $([\alpha ]\ast [\beta ])\ast [\gamma ]=[\alpha ]\ast ([\beta ]\ast [\gamma ])$
(2) $[\alpha ]\ast [c_x]=[\alpha ]$ and $[c_x]\ast [\alpha ]=[\alpha ]$ (unit element)
(3) $[\alpha ]\ast [{\alpha }^{-1}]=[c_x]$ , hence $[\alpha {]}^{-1}=[{\alpha }^{-1}]$ (inverse).

Proof

(1) Let $F(s,t)$ be a homotopy between $(\alpha \ast \beta )\ast \gamma$ and $\alpha \ast (\beta \ast \gamma )$ . It may be given by (figure 4.5(a))

(a)

(b)

Figure 4.5. (a) A homotopy between $(\alpha \ast \beta )\ast \gamma$ and $\alpha \ast (\beta \ast \gamma )$ . (b) A homotopy between $\alpha \ast c_x$ and $\alpha$ .

$$F(s,t)=\{\begin{array}{ll}\alpha \left(\frac{4s}{1+t}\right)& 0\le s\le \frac{1+t}{4}\\ \beta (4s-t-1)& \frac{1+t}{4}\le s\le \frac{2+t}{4}\\ \gamma \left(\frac{4s-t-2}{2-t}\right)& \frac{2+t}{4}\le s\le 1.\end{array}$$

Thus, we may simply write $[\alpha \ast \beta \ast \gamma ]$ to denote $[(a\ast \beta )\ast \gamma ]$ or $[\alpha \ast (\beta \ast \gamma )]$

(2) Define a homotopy $F(s,t)$ by (figure 4.5(b))

$$F(s,t)=\{\begin{array}{ll}\alpha \left(\frac{2s}{1+t}\right)& 0\le s\le \frac{t+1}{2}\\ x& \frac{t+1}{2}\le s\le 1.\end{array}$$

Clearly this is a homotopy between $\alpha \ast c_x$ and $\alpha$ . Similarly, a homotopy between $c_x\ast \alpha$ and $\alpha$ is given by

$$F(s,t)=\{\begin{array}{ll}x& 0\le s\le \frac{1-t}{2}\\ \alpha \left(\frac{2s-1+t}{1+t}\right)& \frac{1-t}{2}\le s\le 1.\end{array}$$

This shows that $[\alpha ]\ast [c_x]=[\alpha ]=[c_x]\ast [\alpha ]$ .

(3) Define a map $F:I\times I\to X$ by

$$F(s,t)=\{\begin{array}{ll}\alpha (2s(1-t))& 0\le s\le \frac{1}{2}\\ \alpha (2(1-s)(1-t))& \frac{1}{2}\le s\le 1.\end{array}$$

Clearly $F(s,0)=\alpha \ast {\alpha }^{-1}$ and $F(s,1)=c_x$ , hence

$$[\alpha \ast {\alpha }^{-1}]=[\alpha ]\ast [{\alpha }^{-1}]=[c_x].$$

This shows that $[{\alpha }^{-1}]=[\alpha {]}^{-1}$ .

In summary, ${\pi }_1(X,x)$ is a group whose unit element is the homotopy class of the constant loop $c_x$ . The product $[\alpha ]\ast [\beta ]$ is well defined and satisfies the

group axioms. The inverse of $[\alpha ]$ is $[\alpha {]}^{-1}=[{\alpha }^{-1}]$ . In the next section we study the general properties of fundamental groups, which simplify the actual computations.

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