The Fundamental Group

Source: Topology (2nd Economy Edition)

If we pick a point $x_0$ of the space X to serve as a “base point” and consider only those paths that begin and end at $x_0$ , the set of these path-homotopy classes does form a group under *. It will be called the fundamental group of X.

The important thing about the fundamental group is that it is a topological invariant of the space X, and it will be crucial in studying homeomorphism problems.

Definition of fundamental group:
Let X be a space; let $x_0$ be a point of X. A path in X that begins and ends at $x_0$ is called a loop based at $x_0$ . The set of path homotopy classes of loops based at $x_0$ , with operation *, is defined as the fundamental group of X relative to the base point $x_0$ . It is denoted by $\pi_1 (X,x_0)$ .

Previously, we have shown that the operation * satisfies the axioms for a group. (See our earlier blog posts on the associativity properties and other groupoid properties of the operation.

This group is called the first homotopy group of X. There is also a second homotopy group, and even groups $\pi_n (X,x_0)$ for all $n\in \mathbb{Z}^+$ .

An example of a fundamental group:

$\pi_1 (\mathbb{R}^n,x_0)$ is the trivial group (the group consisting of just the identity). This is because if f is a loop in $\mathbb{R}^n$ based at $x_0$ , the straight line homotopy is a path homotopy between f and the constant path at $x_0$ .