# Homotopy between composites associated in different ways

## Statement

### Existential version

Suppose $f_1,f_2,f_3$ are loops based at a point $x_0$ in a topological space $X$. We can consider two differently associated products of these three loops:

$\! a = (f_1 * f_2) * f_3, b = f_1 * (f_2 * f_3)$

$a$ and $b$ are homotopic loops, i.e., they are in the same homotopy class of loops based at $x_0$.

This version is essentially the associativity part of showing that the Fundamental group (?) of a based topological space is indeed a group.

### Constructive/explicit version

We first note the explicit piecewise definitions of $a$ and $b$:

$a = \lbrace\begin{array}{rl} f_1(4t), & 0 \le t \le 1/4 \\ f_2(4t - 1), & 1/4 < t \le 1/2 \\ f_3(2t - 1), & 1/2 < t \le 1 \\\end{array}$

and:

$b = \lbrace\begin{array}{rl} f_1(2t), & 0 \le t \le 1/2 \\ f_2(4t - 2), & 1/2 < t \le 3/4 \\ f_3(4t - 3), & 3/4 < t \le 1 \\\end{array}$

If we denote the homotopy by $H$, we want $H(t,0) = a(t), H(t,1) = b(t)$ and $H(0,s) = H(1,s) = x_0$. This homotopy is explicitly given by:

$H(t,s) = \lbrace\begin{array}{rl} f_1\left(\frac{4t}{1 + s}\right), & 0 \le t \le (1 + s)/4 \\ f_2(4t - 1 - s), & (1 + s)/4 < t \le (2 + s)/4 \\ f_3\left(\frac{4t - 2 - s}{2 - s}\right), & (2 + s)/4 < t \le 1 \\\end{array}$

### Uniform version

This version is a little stronger than the other versions. Let $L = \Omega(X,x_0)$ be the loop space of $(X,x_0)$, i.e., the space of all loops in $X$ based at $x_0$ under the compact-open topology. Then, consider the following two maps:

$\! A:L \times L \times L \to L, \qquad A(f_1,f_2,f_3) = (f_1 * f_2) * f_3$

and:

$\! B:L \times L \times L \to L, \qquad B(f_1,f_2,f_3) = f_1 * (f_2 * f_3)$

Then, the maps $A$ and $B$ are homotopic maps. This is part of the proof that $\Omega(X,x_0)$ is a H-space, which is a homotopy variant of topological monoid.