Fundamental Group

The Fundamental Group is an algebraic tool in topology that captures information about loops in a space, measuring its "holes."

Definition

For a space \(X\) with base point \(x_0\), the fundamental group \(\pi_1(X, x_0)\) is the group of homotopy classes of loops based at \(x_0\).

Examples

For a circle \(S^1\), \(\pi_1(S^1) \cong \mathbb{Z}\) (loops winding around), but for a sphere \(S^2\), \(\pi_1(S^2) = 0\) (no holes).

Applications

Used in algebraic topology to classify spaces and in physics to study particle paths.