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.

An interesting question (discussed in the next upcoming blog posts) would be how the group depends on the base point x_0.


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