4.2 General properties of fundamental groups

4.2.1 Arcwise connectedness and fundamental groups

In section 2.3 we defined a topological space $X$ to be arcwise connected if, for any $x_0,x_1\in X$ , there exists a path $\alpha$ such that $\alpha (0)=x_0$ and $\alpha (1)=x_1$ .

Theorem 4.2

Let $X$ be an arcwise connected topological space and let $x_0,x_1\in X$ . Then ${\pi }_1(X,x_0)$ is isomorphic to ${\pi }_1(X,x_1)$ .

Proof

Let $\eta :I\to X$ be a path such that $\eta (0)=x_0$ and $\eta (1)=x_1$ . If $\alpha$ is a loop at $x_0$ , then ${\eta }^{-1}\ast \alpha \ast \eta$ is a loop at $x_1$ (figure 4.6). Given an element $[\alpha ]\in {\pi }_1(X,x_0)$ , this correspondence induces a unique element $[{\alpha }^{\prime }]=[{\eta }^{-1}\ast \alpha \ast \eta ]\in {\pi }_1(X,x_1)$ . We denote this map by $P_{\eta }:{\pi }_1(X,x_0)\to {\pi }_1(X,x_1)$ so that $[{\alpha }^{\prime }]=P_{\eta }([\alpha ])$ .

Figure 4.6. From a loop $\alpha$ at $x_0$ , a loop ${\eta }^{-1}\ast \alpha \ast \eta$ at $x_1$ is constructed.

We show that $P_{\eta }$ is an isomorphism. First, $P_{\eta }$ is a homomorphism, since for $[\alpha ],[\beta ]\in {\pi }_1(X,x_0)$ , we have

$$\begin{array}{rl}{P}_{\eta }([\alpha ]\ast [\beta ])& =[{\eta }^{-1}]\ast [\alpha ]\ast [\beta ]\ast [\eta ]\\ & =[{\eta }^{-1}]\ast [\alpha ]\ast [\eta ]\ast [{\eta }^{-1}]\ast [\beta ]\ast [\eta ]\\ & =P_{\eta }([\alpha ])\ast P_{\eta }([\beta ]).\end{array}$$

To show that $P_{\eta }$ is bijective, we introduce the inverse of $P_{\eta }$ . Define a map $P_{\eta }^{-1}:{\pi }_1(X,x_1)\to {\pi }_1(X,x_0)$ whose action on $[{\alpha }^{\prime }]$ is $P_{\eta }^{-1}([{\alpha }^{\prime }])=[\eta \ast \alpha \ast {\eta }^{-1}]$ .

Clearly $P^{-1}$ is the inverse of $P_{\eta }$ since

$$P_{\eta }^{-1}\circ P_{\eta }([\alpha ])=P_{\eta }^{-1}([{\eta }^{-1}\ast \alpha \ast \eta ])=[\eta \ast {\eta }^{-1}\ast \alpha \ast \eta \ast {\eta }^{-1}]=[\alpha ].$$

Thus, $P_{\eta }^{-1}\circ P_{\eta }={\mathrm{id}}_{{\pi }_1(X,x_0)}$ . From the symmetry, we have $P_{\eta }\circ P_{\eta }^{-1}={\mathrm{id}}_{{\pi }_1(X,x_1)}$ . We find from exercise 2.3 that $P_{\eta }$ is one to one and onto.

Accordingly, if $X$ is arcwise connected, we do not need to specify the base point since ${\pi }_1(X,x_0)\cong {\pi }_1(X,x_1)$ for any $x_0,x_1\in X$ , and we may simply write ${\pi }_1(X)$ .

Exercise 4.1

(1) Let $\eta$ and $\zeta$ be paths from $x_0$ to $x_1$ , such that $\eta \sim \zeta$ . Show that $P_{\eta }=P_{\zeta }$ .

(2) Let $\eta$ and $\zeta$ be paths such that $\eta (1)=\zeta (0)$ . Show that $P_{\eta \ast \zeta }=P_{\zeta }\circ P_{\eta }$ .

4.2.2 Homotopic invariance of fundamental groups

The homotopic equivalence of paths and loops is easily generalized to arbitrary maps. Let $f,g:X\to Y$ be continuous maps. If there exists a continuous map $F:X\times I\to Y$ such that $F(x,0)=f(x)$ and $F(x,1)=g(x)$ , $f$ is said to be homotopic to $g$ , denoted by $f\sim g$ . The map $F$ is called a homotopy between $f$ and $g$ .

Definition 4.6

Let $X$ and $Y$ be topological spaces. $X$ and $Y$ are of the same homotopy type, written as $X\simeq Y$ , if there exist continuous maps $f:X\to Y$ and $g:Y\to X$ such that $f\circ g\sim {\mathrm{id}}_Y$ and $g\circ f\sim {\mathrm{id}}_X$ . The map $f$ is called the homotopy equivalence and $g$ , its homotopy inverse. [Remark: If $X$ is homeomorphic to $Y$ , $X$ and $Y$ are of the same homotopy type but the converse is not necessarily true. For example, a point $\{p\}$ and the real line $\mathbb{R}$ are of the same homotopy type but $\{p\}$ is not homeomorphic to $\mathbb{R}$ .]

Proposition 4.2

‘Of the same homotopy type’ is an equivalence relation in the set of topological spaces.

Proof

Reflectivity: $X\simeq X$ where ${\mathrm{id}}_X$ is a homotopy equivalence. Symmetry: Let $X\simeq Y$ with the homotopy equivalence $f:X\to Y$ . Then $Y\simeq X$ , the homotopy equivalence being the homotopy inverse of $f$ . Transitivity: Let $X\simeq Y$ and $Y\simeq Z$ . Suppose $f:X\to Y$ , $g:Y\to Z$ are homotopy equivalences and $f^{\prime }:Y\to X$ , $g^{\prime }:Z\to Y$ , their homotopy inverses. Then

$$\begin{array}{rl}& (g\circ f)\left(f^{\prime }\circ g^{\prime }\right)=g\left(f\circ f^{\prime }\right)g^{\prime }\sim g\circ {\mathrm{id}}_Y\circ g^{\prime }=g\circ g^{\prime }\sim {\mathrm{id}}_Z\\ & (f^{\prime }\circ g^{\prime })(g\circ f)=f^{\prime }(g^{\prime }\circ g)f\sim f^{\prime }\circ {\mathrm{id}}_Y\circ f=f^{\prime }\circ f\sim {\mathrm{id}}_X\end{array}$$

from which it follows $X\simeq Z$

$\square$

One of the most remarkable properties of the fundamental groups is that two topological spaces of the same homotopy type have the same fundamental group.

Theorem 4.3

Let $X$ and $Y$ be topological spaces of the same homotopy type. If $f:X\to Y$ is a homotopy equivalence, ${\pi }_1(X,x_0)$ is isomorphic to ${\pi }_1(Y,f(x_0))$ .

The following corollary follows directly from theorem 4.3.

Corollary 4.1

A fundamental group is invariant under homeomorphisms, and hence is a topological invariant.

In this sense, we must admit that fundamental groups classify topological spaces in a less strict manner than homeomorphisms. What we claim at most is that if topological spaces $X$ and $Y$ have different fundamental groups, $X$ cannot be homeomorphic to $Y$ . Note, however, that the homotopy groups including the fundamental groups have many applications to physics as we shall see in due course. We should stress that the main usage of the homotopy groups in physics is not to classify spaces but to classify maps or field configurations.

It is rather difficult to appreciate what is meant by ‘of the same homotopy type’ for an arbitrary pair of $X$ and $Y$ . In practice, however, it often happens that $Y$ is a subspace of $X$ . We then claim that $X\simeq Y$ if $Y$ is obtained by a continuous deformation of $X$ .

Definition 4.7

Let $R$ ( $\ne \varnothing$ ) be a subspace of $X$ . If there exists a continuous map $f:X\to R$ such that $f{|}_R={\mathrm{id}}_R$ , $R$ is called a retract of $X$ and $f$ a retraction.

Note that the whole of $X$ is mapped onto $R$ keeping points in $R$ fixed. Figure 4.7 is an example of a retract and retraction.

Figure 4.7. The circle $R$ is a retract of the annulus $X$ . The arrows depict the action of the retraction.

Definition 4.8

Let $R$ be a subspace of $X$ . If there exists a continuous map $H:X\times I\to X$ such that

$$H(x,0)=xH(x,1)\in R\text{f o r a n y}x\in X\text{\hspace{1em}(4.5)}$$

$$H(x,t)=x\text{f o r a n y}x\in R\text{a n d a n y}t\in I.\text{\hspace{1em}(4.6)}$$

The space $R$ is said to be a deformation retract of $X$ . Note that $H$ is a homotopy between ${\mathrm{id}}_X$ and a retraction $f:X\to R$ , which leaves all the points in $R$ fixed during deformation.

A retract is not necessarily a deformation retract. In figure 4.8, the circle $R$ is a retract of $X$ but not a deformation retract, since the hole in $X$ is an obstruction to continuous deformation of ${\mathrm{id}}_X$ to the retraction. Since $X$ and $R$ are of the same homotopy type, we have

Figure 4.8. The circle $R$ is not a deformation retract of $X$ .

$${\pi }_1(X,a)\cong {\pi }_1(R,a)a\in R.\text{\hspace{1em}(4.7)}$$

Example 4.1

Let $X$ be the unit circle and $Y$ be the annulus,

$$X=\left\{{\mathrm{e}}^{\mathrm{i}\theta }\mid 0\le \theta <2\pi \right\}\text{\hspace{1em}(4.8)}$$

$$Y=\left\{r{e}^{i\theta }\mid 0\le \theta <2\pi ,\frac{1}{2}\le r\le \frac{2}{3}\right\}\text{\hspace{1em}(4.9)}$$

see figure 4.7. Define $f:X\hookrightarrow Y$ by $f({\mathrm{e}}^{\mathrm{i}\theta })={\mathrm{e}}^{\mathrm{i}\theta }$ and $g:Y\to X$ by $g(r{e}^{\mathrm{i}\theta })={\mathrm{e}}^{\mathrm{i}\theta }$ . Then $f\circ g:r{e}^{\mathrm{i}\theta }\mapsto {\mathrm{e}}^{\mathrm{i}\theta }$ and $g\circ f:{\mathrm{e}}^{\mathrm{i}\theta }\mapsto {\mathrm{e}}^{\mathrm{i}\theta }$ . Observe that $f\circ g\sim {\mathrm{id}}_Y$ and $g\circ f={\mathrm{id}}_X$ . There exists a homotopy

$$H(r{\mathrm{e}}^{\mathrm{i}\theta },t)=\{1+(r-1)(1-t)\}{\mathrm{e}}^{\mathrm{i}\theta }$$

which interpolates between ${\mathrm{id}}_X$ and $f\circ g$ , keeping the points on $X$ fixed. Hence, $X$ is a deformation retract of $Y$ . As for the fundamental groups we have ${\pi }_1(X,a)\cong {\pi }_1(Y,a)$ where $a\in X$ .

Definition 4.9

If a point $a\in X$ is a deformation retract of $X$ , $X$ is said to be contractible.

Let $c_a:X\to \{a\}$ be a constant map. If $X$ is contractible, there exists a homotopy $H:X\times I\to X$ such that $H(x,0)=c_a(x)=a$ and $H(x,1)={\mathrm{id}}_X(x)=x$ for any $x\in X$ and, moreover, $H(a,t)=a$ for any $t\in I$ . The homotopy $H$ is called the contraction.

Example 4.2

$X={\mathbb{R}}^n$ is contractible to the origin 0. In fact, if we define $H:{\mathbb{R}}^n\times I\to \mathbb{R}$ by $H(x,t)=tx$ , we have (i) $H(x,0)=0$ and $H(x,1)=x$ for any $x\in X$ and (ii) $H(0,1)=0$ for any $t\in I$ . Now it is clear that any convex subset of ${\mathbb{R}}^n$ is contractible.

Exercise 4.2

Let $D^2=\{(x,y)\in {\mathbb{R}}^2|x^2+y^2\le 1\}$ . Show that the unit circle $S^1$ is a deformation retract of $D^2-\{0\}$ . Show also that the unit sphere $S^n$ is a deformation retract of $D^{n+1}-\{0\}$ , where $D^{n+1}=\{x\in {\mathbb{R}}^{n+1}||x|\le 1\}$ .

Theorem 4.4

The fundamental group of a contractible space $X$ is trivial, ${\pi }_1(X,x_0)\cong \{e\}$ . In particular, the fundamental group of ${\mathbb{R}}^n$ is trivial, ${\pi }_1({\mathbb{R}}^n,x_0)\cong \{e\}$ .

Proof

A contractible space has the same fundamental group as a point $\{p\}$ and a point has a trivial fundamental group.

If an arcwise connected space $X$ has a trivial fundamental group, $X$ is said to be simply connected, see section 2.3.

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