Homotopy between composites of homotopic loops

Existential version
Suppose $$X$$ is a topological space, $$x_0$$ is a point in $$X$$, and $$f_1,g_1,f_2,g_2$$ are loops based at $$x_0$$ with the property that $$f_1$$ is homotopic to $$g_1$$ (as a loop based at $$x_0$$) and $$f_2$$ is homotopic to $$g_2$$ (again, as a loop based at $$x_0$$). Then, $$f_1 * f_2$$ is homotopic to $$g_1 * g_2$$.

Constructive/explicit version
More explicitly, suppose $$F_1$$ is a homotopy from $$f_1$$ to $$g_1$$. In other words, $$F_1:S^1 \times I \to X$$ is a continuous map (where $$S^1$$ is the circle, viewed as $$[0,1]$$ with endpoints identified, and $$I = [0,1]$$ is the closed unit interval) having the following properties:


 * $$F_1(s,0) = f_1(s)$$
 * $$F_1(s,1) = g_1(s)$$
 * $$F_1(0,t) = x_0$$ (here $$\! 0 \sim 1$$ is the chosen basepoint of the circle from which we're mapping). This says that the loop always remains based on $$x_0$$.

Similarly, suppose $$F_2:S^1 \times I \to X$$ is a continuous map having the following properties:


 * $$F_2(s,0) = f_2(s)$$
 * $$F_2(s,1) = g_2(s)$$
 * $$F_2(0,t) = x_0$$ (here $$\! 0 \sim 1$$ is the chosen basepoint of the circle). This says that the loop always remains based on $$x_0$$.

Then, we can consider the following homotopy from $$f_1 * f_2$$ to $$g_1 * g_2$$:

$$F(s,t) := \lbrace\begin{array}{rl} F_1(2s,t), & 0 \le t \le 1/2 \\ F_2(2s-1,t), & 1/2 < t \le 1 \\\end{array}$$

We can think of $$F$$ as $$F_1 * F_2$$.

Graphical version
The pictures below describe the explicit construction. Note that the geometric shapes shown in these pictures can be thought of as the sources of the respective maps to $$X$$, with the additional caveat that the boundary vertical lines map to the point $$x_0$$. (The same pictures, without the collapse of boundaries, work to establish the homotopy between composites of homotopic paths).

The homotopy $$F_1$$ between $$f_1$$ and $$g_1$$ is a map from a filled unit square, where the restrictions of the map to the bottom and top sides of the square are $$f_1$$ and $$g_1$$ respectively. The left and right sides map to the point $$x_0$$:



The homotopy $$F_2$$ between $$f_2$$ and $$g_2$$ is a map from a filled unit square, where the restrictions of the map to the bottom and top sides of the square are $$f_2$$ and $$g_2$$ respectively. The left and right sides map to the point $$x_0$$:



These homotopies are composed by concatenation, as shown below. Both $$F_1$$ and $$F_2$$ need to be scaled by a factor of $$1/2$$ for the concatenated homotopy to fit in a unit square: