# Homotopy between composites of homotopic loops

## Statement

### 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: