Geometric Topology 17

The Fundamental Group

Fix a "basepoint" $x_0 \in X$ . Recall that a loop at $x_0$ is a path from $x_0$ to $x_0$ . For example, the constant loop $\varepsilon$ .

Definition: $\pi_1 (X,x_0) = \{[\alpha] : \alpha \ \text{is a loop based at } x_0\}$ . The product of two elements of $\pi_1(X,x_0)$ is given by $[\alpha][\beta] = [\alpha \beta]$ .

We shall show that this definition is well-defined, and makes $\pi_1 (X,x_0)$ into a group with identity element $e = [\varepsilon]$ , and inverses $[\alpha]^{-1} = [\alpha^{-1}]$ .

Lemma 1: If $\alpha \approxeq_F \alpha '$ and $\beta \approxeq_G \beta '$ then $\alpha \beta \approxeq \alpha ' \beta '$ .

Proof: Take:

H(s,t) = \begin{cases} F(2s,t) & \ \text{if} \ s \in [0,\frac{1}{2}] \\ G(2s-1,t) & \ \text{if} \ s \in [\frac{1}{2},1], \end{cases}

noting that $F(1,t) = x_0 = G(1,t)$ . The map $H$ is therefore continuous, as in the proof of Lemma 0. It gives the required path-homotopy $\alpha \beta \approxeq \alpha ' \beta '$ . $\blacksquare$

Group Axioms

Lemma 2: $\varepsilon \alpha \approxeq \alpha \approxeq \alpha \varepsilon$ . Lemma 3: $\alpha \alpha^{-1} \approxeq \varepsilon \approxeq \alpha^{-1}\alpha$ . Lemma 4: (\alpha \beta)\gamma \approxeq_H \alpha(\beta \gamma).

Proof:

H(s,t) = \begin{cases} \alpha(\frac{4s}{1+t}) & \text{if} \ s \leq \frac{1}{4}(1+t) \\ \beta(4s - 1 - t) & \text{if} \ \frac{1}{4}(1+t) \leq s \leq \frac{1}{4}(2+t) \\ \gamma (\frac{4s-2-t}{2-t}) & \text{if} \ \frac{1}{4}(2+t) \leq s. \end{cases}

The Clifford Torus

We are used to thinking of the torus as embedded as the surface of a doughnut in $\mathbb{R}^3$ , but it is more naturally the subset:

\{(\cos(2\pi s), \sin(2\pi s), \cos(2\pi t), \sin(2\pi t)): (s,t) \in I^2\} \subset S^3 \subset \mathbb{R}^4

As an abstract topological space, we know that $T = \hat{I}^2$ is the quotient of the unit square by means of the word $aba^{-1}b^{-1}$ . With this description, $a$ and $b$ define loops in $T$ based at the unique vertex $x_0 \in T$ .

The picture describes a homotopy $aba^{-1}b^{-1} \approxeq \varepsilon$ . Thus $[a][b][a]^{-1}[b]^{-1} = e$ , and:

[a][b] = [b][a]

in $\pi_1(T,x_0)$ . The latter is an abelian group isomorphic to $\mathbb{Z}^2 = \mathbb{Z} \times \mathbb{Z}$ , and $T$ is a quotient group:

\mathbb{R}^2 / \mathbb{Z}^2 \cong (\mathbb{R}/\mathbb{Z}) \times (\mathbb{R}/\mathbb{Z}) \cong S^1 \times S^1.

Properties of $\pi_1$

Suppose that $X$ is a path-connected topological space. Let $x_0,x_1 \in X$ , and choose a path $\sigma$ from $x_0$ to $x_1$ . Define a map $\phi_{\sigma} : \pi_1 (X,x_0) \rightarrow \pi_1 (X,x_1)$ by:

\phi_{\sigma} [\alpha] = [\sigma^{-1}\alpha \sigma].

It is well-defined.

Lemma 5: $\alpha \approxeq \alpha '$ implies that $\sigma^{-1}\alpha \sigma \approxeq \sigma^{-1} \alpha ' \sigma$ . The map is also a group homomorphism:

Lemma 6: $\sigma^{-1}(\alpha \beta) \sigma \approxeq (\sigma^{-1}\alpha \sigma)(\sigma^{-1}\beta \sigma).$

Lemma 7: $\phi_{\sigma}$ is bijective.

It follows that $\pi_1(X,x_0) \cong \pi_1(X,x_1)$ as abstract groups, so we can speak of 'the' fundamental group $\pi_1(X)$ .

Functoriality

Suppose that $X,Y$ are topological spaces and $f: X \rightarrow Y$ is continuous.

Definition: The induced mapping on fundamental groups is:

f_{*} : \pi_1 (X,x_0) \rightarrow \pi_1 (Y,y_0), \ \text{where} \ y_0 = f(x_0),

given by $f_{*}[\alpha] = [f \circ \alpha]$ . $f_{*}$ is well-defined, because we can compose a homotopy $H_t : \alpha \approxeq \alpha '$ by applying $f$ to it.

Lemma 8: $f_{*}$ is a group homomorphism.

Proof: If $\alpha,\beta$ are loops in $X$ based at $x_0$ then:

f_{*}([\alpha][\beta]) = f_{*}([\alpha \beta]) = [f \circ (\alpha \beta)] = [(f \circ \alpha)(f \circ \beta)] = (f_{*}[\alpha])(f_{*}[\beta]).

It is equally obvious that $1_{*} = 1$ and $(g \circ f)_{*} = g_{*} \circ f_{*}$ .

If $f$ is a homeomorphism, we can take $g = f^{-1}$ to deduce that $f_{*}$ is an isomorphism of groups:

Corollary: If $X$ is homeomorphic to $Y$ (often written $X \approx Y$ ) then $\pi_1(X) \cong \pi_1(Y)$ .

Simple Spaces

Definition: A topological space $X$ is simply-connected if it is path-connected and $\pi_1(X) = \{e\}$ is the trivial group.

An obvious example is $\mathbb{R}^n$ . Take $x_0$ as the origin. Given a loop $\alpha$ based at $x_0$ , we can define $H(s,t) = \alpha (s)t$ in order to 'shrink' $\alpha$ to a point. So $\alpha \approxeq \varepsilon$ and $[\alpha] = e$ .

In fact $\mathbb{R}^n$ has the much stronger property of being contractible, meaning that the whole space is homotopic to a point, i.e. there exists $x_0 \in X$ and a map $H: X \times I \rightarrow X$ such that:

H(x,t) = \begin{cases} x & \text{if} \ t = 0 \\ x_0 & \text{if} \ t = 1. \end{cases}

More examples: A star-shaped domain in $\mathbb{C}$ , a tree (a connected graph with no cycles).

There are many spaces that are simply-connected but not contractible, such as $S^n$ for $n \geq 2$ . For simplicity, consider $S = S^2$ ; the open set $S^2 - \{n\}$ (where $n = (0,0,1)$ ) is homeomorphic to $\mathbb{R}^2$ via stereographic projection, and any loop $\alpha$ (based at say $s$ ) can be deformed so as to avoid $n$ , hence shrunk to a point.