Homotopy between composites of homotopic loops

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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:

Homotopyleftofcomposition.png

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:

Homotopyrightofcomposition.png

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:

Homotopyofcompositepaths.png