Fundamental group

Summary

The fundamental group $\pi_1(X, x_0)$ of a pointed space is the group of loops at $x_0$ up to path homotopy, with concatenation of loops as product. It is the first invariant of algebraic topology that detects one-dimensional holes. Homotopy equivalent path-connected spaces have isomorphic fundamental groups. For example, the plane has trivial fundamental group, while the circle has fundamental group $\Z$.