Paths on a surface 
Associativity
When we say that the product of two real numbers is associative we are refering to the property \((a.b).c=a.(b.c)\). That is, in the product of three real numbers, it is irrelevant to start by the product of the first two factors and then multiply by the last factor, or to start by the product of the last two factors and then multiply the first factor.
Considering homotopy classes, to check associativity on the product of paths we need to verify the same property: that given three paths, the path resulting from the product of the first two followed by the third is homotopic to the path resulting from the product of the first by the product of the last two.
More exactly, given \(f_{1}\), \(f_{2}\) and \(f_{3}\) from \([0,1]\) to \(S\), such that \(f_{1}(1)=f_{2}(0)\) and \(f_{2}(1)=f_{3}(0)\), \((f_{1}.f_{2}).f_{3}\) is homotopic to \(f_{1}.(f_{2}.f_{3})\).
Given that we consider paths covered in one hour, the products above will differ in time and velocity with which we cover each path.
On the first case, \((f_{1}.f_{2}).f_{3}\):
The final product is \((f_{1}.f_{2})\) by \(f_{3}\), that is, the first half of an hour for \((f_{1}.f_{2})\) and the last half of an hour for \(f_{3}\); as in the first half of an hour we have to cover two paths, the first quarter of an hour is for \(f_{1}\) and the second for \(f_{2}\). That is, in the first quarter of an hour we cover \(f_{1}\) (at a velocity four times higher than on the original path), in the second quarter of an hour we cover \(f_{2}\) (also at a velocity four times higher) and in the last half an hour we cover \(f_{3}\) (at twice the speed of the original path).
Formaly, \(g_{1}=(f_{1}.f_{2}).f_{3}\) from \([0,1]\) to \(S\) can be defined as follows: \[g_{1}(t)=\begin{cases} \begin{array}{lcl} f_{1}(4t) & \mbox{if} & 0\leq t\leq0.25\\ f_{2}(4t-1) & \mbox{if} & 0.25\leq t\leq0.5\\ f_{3}(2t-1) & \mbox{if} & 0.5\leq t\leq1 \end{array}\end{cases}\]
On the second case, \(f_{1}.(f_{2}.f_{3})\):
The final product is the product of \(f_{1}\) by \((f_{2}.f_{3})\), that is, the first half an hour for \(f_{1}\) and the last half an hour for \((f_{2}.f_{3})\); as in the first half an hour we have to cover two paths, we choose a quarter of an hour for each path. That is, in the first half an hour we cover \(f_{1}\)(at twice the velocity of the original path), in the third quarter of an hour we cover \(f_{2}\) (at four times the velocity of the original path) and in the last quarter of an hour we cover \(f_{3}\) (also at four times the velocity of the original path).
Formaly, \(g_{2}=f_{1}.(f_{2}.f_{3})\) from \([0,1]\) to \(S\) can be defined as follows:\[g_{2}(t)=\begin{cases} \begin{array}{lcl} f_{1}(2t) & \mbox{if} & 0\leq t\leq0.5\\ f_{2}(4t-2) & \mbox{if} & 0.5\leq t\leq0.75\\ f_{3}(4t-3) & \mbox{if} & 0.75\leq t\leq1 \end{array}\end{cases}\]
The paths are different! However, associativity is valid on the homotopy classes, as it can be shown that the two resulting paths are homotopic, it is only needed a continuous adjustment on the velocities of the paths (or on the coloring of the corresponding rubber bands).
Formaly, one homotopy \(H\) from \([0,1]\times[0,1]\) to \(S\) (between \(g_{2}\) and \(g_{1}\)) can be defined as follows:\[H(t,x)=\begin{cases} \begin{array}{lcl} f_{1}\left(\frac{4t}{2-x}\right) & \mbox{if} & 0\leq t\leq\frac{2-x}{4}\\ f_{2}(4t-2+x) & \mbox{if} & \frac{2-x}{4}<t\leq\frac{3-x}{4}\\ f_{3}\left(\frac{-3+4t+x}{1+x}\right) & \mbox{if} & \frac{3-x}{4}<t\leq1 \end{array}\end{cases}\]
