Homotopy between composites of homotopic loops: Difference between revisions

Latest revision as of 18:45, 18 December 2010

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 $g_{1} * g_{2}$ :

$F (s, t) : = {\begin{matrix} F_{1} (2 s, t), & 0 \leq t \leq 1 / 2 \\ F_{2} (2 s - 1, t), & 1 / 2 < t \leq 1 \end{matrix}$

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:

@@ Line 1: / Line 1: @@
 ==Statement==
+===Existential version===
 Suppose <math>X</math> is a [[topological space]], <math>x_0</math> is a point in <math>X</math>, and <math>f_1,g_1,f_2,g_2</math> are loops based at <math>x_0</math> with the property that <math>f_1</math> is homotopic to <math>g_1</math> (as a loop based at <math>x_0</math>) and <math>f_2</math> is homotopic to <math>g_2</math> (again, as a loop based at <math>x_0</math>). Then, <math>f_1 * f_2</math> is homotopic to <math>g_1 * g_2</math>.
+===Constructive/explicit version===
+More explicitly, suppose <math>F_1</math> is a homotopy from <math>f_1</math> to <math>g_1</math>. In other words, <math>F_1:S^1 \times I \to X</math> is a continuous map (where <math>S^1</math> is the [[circle]], viewed as <math>[0,1]</math> with endpoints identified, and <math>I = [0,1]</math> is the [[closed unit interval]]) having the following properties:
+* <math>F_1(s,0) = f_1(s)</math>
+* <math>F_1(s,1) = g_1(s)</math>
+* <math>F_1(0,t) = x_0</math> (here <math>\! 0 \sim 1</math> is the chosen basepoint of the circle from which we're mapping). This says that the loop always remains based on <math>x_0</math>.
+Similarly, suppose <math>F_2:S^1 \times I \to X</math> is a continuous map having the following properties:
+* <math>F_2(s,0) = f_2(s)</math>
+* <math>F_2(s,1) = g_2(s)</math>
+* <math>F_2(0,t) = x_0</math> (here <math>\! 0 \sim 1</math> is the chosen basepoint of the circle). This says that the loop always remains based on <math>x_0</math>.
+Then, we can consider the following homotopy from <math>f_1 * f_2</math> to <math>g_1 * g_2</math>:
+<math>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}</math>
+We can think of <math>F</math> as <math>F_1 * F_2</math>.
+===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 <math>X</math>, with the additional caveat that the boundary vertical lines map to the point <math>x_0</math>. (The same pictures, without the collapse of boundaries, work to establish the [[homotopy between composites of homotopic paths]]).
+The homotopy <math>F_1</math> between <math>f_1</math> and <math>g_1</math> is a map from a filled unit square, where the restrictions of the map to the bottom and top sides of the square are <math>f_1</math> and <math>g_1</math> respectively. The left and right sides map to the point <math>x_0</math>:
+[[File:Homotopyleftofcomposition.png|250px]]
+The homotopy <math>F_2</math> between <math>f_2</math> and <math>g_2</math> is a map from a filled unit square, where the restrictions of the map to the bottom and top sides of the square are <math>f_2</math> and <math>g_2</math> respectively. The left and right sides map to the point <math>x_0</math>:
+[[File:Homotopyrightofcomposition.png|250px]]
+These homotopies are composed by concatenation, as shown below. Both <math>F_1</math> and <math>F_2</math> need to be scaled by a factor of <math>1/2</math> for the concatenated homotopy to fit in a unit square:
+[[File:Homotopyofcompositepaths.png|250px]]