Henri Poincare (1854-1912) discovered the fundamental group (also known as the Poincare group or the first homotopy group) of a manifold (or more generally, any topological space) and used it to classify manifolds. A manifold is a topological space every point of which has a neighborhood which looks like (i.e. is homeomorphic to) n-dimensional real space R^n for some number n. For example, n-dimensional real space, circles, spheres, the surface of a donut, the universe and the complement of a knot (a knot is the mathematical analogue of a loop of string with its ends glued together) in three dimensional space are all manifolds.

If p is a point in a manifold M and c and c' are paths (i.e. continuous maps of [0,1] into M) which begin and end at p then Poincare considered them to be equivalent (i.e. in the same...