The Fundamental Group - Key Properties

The fundamental group possesses remarkable algebraic and topological properties that make it a powerful tool for classifying and distinguishing topological spaces.

Definition

A topological space $X$ is simply connected if it is path-connected and $\pi_1(X, x_0) = \{e\}$ for any (equivalently, for some) basepoint $x_0 \in X$ .

Simple connectivity means every loop can be continuously contracted to a point. The sphere $S^n$ for $n \geq 2$ , the disk $D^n$ for any $n \geq 1$ , and Euclidean space $\mathbb{R}^n$ are all simply connected.

Definition

Let $f : (X, x_0) \to (Y, y_0)$ be a continuous map between pointed spaces. The induced homomorphism $f_* : \pi_1(X, x_0) \to \pi_1(Y, y_0)$ is defined by $f_*([\gamma]) = [f \circ \gamma]$ for any loop $\gamma$ based at $x_0$ .

The induced homomorphism is well-defined: if $\gamma \simeq \gamma'$ via homotopy $H$ , then $f \circ \gamma \simeq f \circ \gamma'$ via the homotopy $f \circ H$ . Moreover, $f_*$ is a group homomorphism: $f_*([\gamma] \cdot [\delta]) = f_*([\gamma]) \cdot f_*([\delta])$ .

Theorem

The fundamental group is a functor from the category of pointed topological spaces to the category of groups:

If $\text{id}_X : X \to X$ is the identity map, then $(\text{id}_X)_* = \text{id}_{\pi_1(X,x_0)}$
If $f : X \to Y$ and $g : Y \to Z$ are continuous, then $(g \circ f)_* = g_* \circ f_*$

This functoriality is fundamental: topologically equivalent spaces have algebraically equivalent fundamental groups. Specifically, if $f : X \to Y$ is a homeomorphism, then $f_* : \pi_1(X, x_0) \to \pi_1(Y, f(x_0))$ is a group isomorphism.

Example

Consider the inclusion $i : S^1 \to \mathbb{R}^2 \setminus \{0\}$ . The induced map $i_* : \pi_1(S^1) \to \pi_1(\mathbb{R}^2 \setminus \{0\})$ is an isomorphism. This reflects the fact that $S^1$ is a deformation retract of the punctured plane.

Definition

Let $X$ be path-connected and $x_0, x_1 \in X$ . For any path $\alpha$ from $x_0$ to $x_1$ , the change of basepoint isomorphism is given by $\beta_\alpha : \pi_1(X, x_0) \to \pi_1(X, x_1), \quad [\gamma] \mapsto [\overline{\alpha} * \gamma * \alpha]$

The map $\beta_\alpha$ is indeed an isomorphism, showing that the fundamental group of a path-connected space is independent of basepoint up to isomorphism. However, the isomorphism itself depends on the choice of path $\alpha$ .

Remark

The basepoint change isomorphism is not canonical unless $\pi_1(X, x_0)$ is abelian. Different paths $\alpha$ and $\alpha'$ from $x_0$ to $x_1$ yield isomorphisms differing by an inner automorphism of $\pi_1(X, x_1)$ .

When working with path-connected spaces, we often write $\pi_1(X)$ without specifying a basepoint, understanding that all choices yield isomorphic groups. For multiply connected spaces, the dependence on basepoint becomes essential to the theory.