The Fundamental Group - Core Definitions

The fundamental group is one of the most important algebraic invariants in topology, providing a way to classify topological spaces by studying loops within them.

Definition

Let $X$ be a topological space and $x_0 \in X$ a basepoint. A path in $X$ from $x_0$ to $x_1$ is a continuous map $\gamma : [0,1] \to X$ with $\gamma(0) = x_0$ and $\gamma(1) = x_1$ . A loop based at $x_0$ is a path with $\gamma(0) = \gamma(1) = x_0$ .

The space of all paths connecting two points carries rich topological structure. We study this structure through the equivalence relation of homotopy, which captures the intuitive notion of "continuously deforming" one path into another.

Definition

Two paths $\gamma_0, \gamma_1 : [0,1] \to X$ from $x_0$ to $x_1$ are path homotopic, written $\gamma_0 \simeq \gamma_1$ , if there exists a continuous map $H : [0,1] \times [0,1] \to X$ such that:

$H(s,0) = \gamma_0(s)$ and $H(s,1) = \gamma_1(s)$ for all $s \in [0,1]$
$H(0,t) = x_0$ and $H(1,t) = x_1$ for all $t \in [0,1]$

The map $H$ is called a path homotopy from $\gamma_0$ to $\gamma_1$ .

Path homotopy defines an equivalence relation on the set of loops based at $x_0$ . The equivalence class of a loop $\gamma$ is denoted $[\gamma]$ . This construction allows us to define the fundamental group.

Definition

Let $X$ be a topological space and $x_0 \in X$ . The fundamental group $\pi_1(X, x_0)$ is the set of homotopy classes of loops based at $x_0$ , equipped with the operation of path concatenation: $[\gamma] \cdot [\delta] = [\gamma * \delta]$ where $(\gamma * \delta)(s) = \begin{cases} \gamma(2s) & 0 \leq s \leq 1/2 \\ \delta(2s-1) & 1/2 \leq s \leq 1 \end{cases}$

Example

For the unit circle $S^1$ with basepoint $(1,0)$ , the fundamental group $\pi_1(S^1) \cong \mathbb{Z}$ . The integer $n$ corresponds to the homotopy class of the loop that winds around the circle $n$ times counterclockwise (negative for clockwise).

The group operation on $\pi_1(X, x_0)$ is well-defined on homotopy classes: if $\gamma_0 \simeq \gamma_1$ and $\delta_0 \simeq \delta_1$ , then $\gamma_0 * \delta_0 \simeq \gamma_1 * \delta_1$ . This makes $\pi_1(X, x_0)$ a genuine group.

Remark

The choice of basepoint matters: for non-path-connected spaces, different basepoints may yield non-isomorphic fundamental groups. However, for path-connected spaces, all choices of basepoint give isomorphic groups.

The constant loop $e_{x_0}(s) = x_0$ serves as the identity element in $\pi_1(X, x_0)$ . The inverse of $[\gamma]$ is given by the reverse path $\overline{\gamma}(s) = \gamma(1-s)$ . These properties establish $\pi_1(X, x_0)$ as a group, not just a set with a binary operation.