Editing Path (topology) (section)

== Path composition ==

One can compose paths in a topological space in the following manner. Suppose <math>f</math> is a path from <math>x</math> to <math>y</math> and <math>g</math> is a path from <math>y</math> to <math>z</math>. The path <math>fg</math> is defined as the path obtained by first traversing <math>f</math> and then traversing <math>g</math>:
:<math>fg(s) = \begin{cases}f(2s) & 0 \leq s \leq \frac{1}{2} \\ g(2s-1) & \frac{1}{2} \leq s \leq 1.\end{cases}</math>
Clearly path composition is only defined when the terminal point of <math>f</math> coincides with the initial point of <math>g.</math> If one considers all loops based at a point <math>x_0,</math> then path composition is a [[binary operation]]. 

Path composition, whenever defined, is not [[associative]] due to the difference in parametrization. However it {{em|is}} associative up to path-homotopy. That is, <math>[(fg)h] = [f(gh)].</math> Path composition defines a [[Group (mathematics)|group structure]] on the set of homotopy classes of loops based at a point <math>x_0</math> in <math>X.</math> The resultant group is called the [[fundamental group]] of <math>X</math> based at <math>x_0,</math> usually denoted <math>\pi_1\left(X, x_0\right).</math> 

In situations calling for associativity of path composition "on the nose," a path in <math>X</math> may instead be defined as a continuous map from an interval <math>[0, a]</math> to <math>X</math> for any real <math>a \geq 0.</math> (Such a path is called a [[Moore path]].) A path <math>f</math> of this kind has a length <math>|f|</math> defined as <math>a.</math>  Path composition is then defined as before with the following modification:
:<math>fg(s) = \begin{cases}f(s) & 0 \leq s \leq |f| \\ g(s-|f|) & |f| \leq s \leq |f| + |g|\end{cases}</math>

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