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 $x_0$.

Constructive/explicit version

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

$a=\{\begin{array}{cc}{f}_1(4t),& 0\le t\le 1/4\\ f_2(4t-1),& 1/4<t\le 1/2\\ f_3(2t-1),& 1/2<t\le 1\\ \end{array}$

and:

$b=\{\begin{array}{cc}{f}_1(2t),& 0\le t\le 1/2\\ f_2(4t-2),& 1/2<t\le 3/4\\ f_3(4t-3),& 3/4<t\le 1\\ \end{array}$

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:

$H(t,s)=\{\begin{array}{cc}{f}_1(2(1+s)t),& 0\le t\le (1+s)/4\\ f_2(4t-1-s),& (1+s)/4<t\le (2+s)/4\\ f_3(2t-1+s(2t-2)),& (2+s)/4<t\le 1\\ \end{array}$

Graphical version

Uniform version

This version is a little stronger than the other versions. Let $L=\mathrm{\Omega }(X,x_0)$ 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:

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

and:

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

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