Path (topology) - Path Composition

Path Composition

One can compose paths in a topological space in an obvious manner. Suppose f is a path from x to y and g is a path from y to z. The path fg is defined as the path obtained by first traversing f and then traversing g:

Clearly path composition is only defined when the terminal point of f coincides with the initial point of g. If one considers all loops based at a point x₀, then path composition is a binary operation.

Path composition, whenever defined, is not associative due to the difference in parametrization. However it is associative up to path-homotopy. That is, = . Path composition defines a group structure on the set of homotopy classes of loops based at a point x₀ in X. The resultant group is called the fundamental group of X based at x₀, usually denoted π₁(X,x₀).

In situations calling for associativity of path composition "on the nose," a path in X may instead be defined as a continuous map from an interval to X for any real a ≥ 0. A path f of this kind has a length |f| defined as a. Path composition is then defined as before with the following modification:

Whereas with the previous definition, f, g, and fg all have length 1 (the length of the domain of the map), this definition makes |fg| = |f| + |g|. What made associativity fail for the previous definition is that although (fg)h and f(gh) have the same length, namely 1, the midpoint of (fg)h occurred between g and h, whereas the midpoint of f(gh) occurred between f and g. With this modified definition (fg)h and f(gh) have the same length, namely |f|+|g|+|h|, and the same midpoint, found at (|f|+|g|+|h|)/2 in both (fg)h and f(gh); more generally they have the same parametrization throughout.