Fundamental Group - Definition

Definition

Let X be a topological space, and let x₀ be a point of X. We are interested in the following set of continuous functions called loops with base point x₀.

Now the fundamental group of X with base point x is this set modulo homotopy

equipped with the group multiplication defined by (f ∗ g)(t) := f(2t) if 0 ≤ t ≤ 1/2 and (f ∗ g)(t) := g(2t − 1) if 1/2 ≤ t ≤ 1. Thus the loop f ∗ g first follows the loop f with "twice the speed" and then follows g with twice the speed. The product of two homotopy classes of loops and is then defined as, and it can be shown that this product does not depend on the choice of representatives.

With the above product, the set of all homotopy classes of loops with base point x₀ forms the fundamental group of X at the point x₀ and is denoted

or simply π(X, x₀). The identity element is the constant map at the basepoint, and the inverse of a loop f is the loop g defined by g(t) = f(1 − t). That is, g follows f backwards.

Although the fundamental group in general depends on the choice of base point, it turns out that, up to isomorphism (actually, even up to inner isomorphism), this choice makes no difference as long as the space X is path-connected. For path-connected spaces, therefore, we can write π₁(X) instead of π₁(X, x₀) without ambiguity whenever we care about the isomorphism class only.