Homotopy between composites associated in different ways: Difference between revisions

Latest revision as of 22:08, 8 January 2011

Statement

Existential version

Suppose $f_{1}, f_{2}, f_{3}$ are loops based at a point $x_{0}$ in a topological space $X$ . We can consider two differently associated products of these three loops:

$a = (f_{1} * f_{2}) * f_{3}, b = f_{1} * (f_{2} * f_{3})$

$a$ and $b$ are homotopic loops, i.e., they are in the same homotopy class of loops based at $x_{0}$ .

This version is essentially the associativity part of showing that the Fundamental group (?) of a based topological space is indeed a group.

Constructive/explicit version

We first note the explicit piecewise definitions of $a$ and $b$ :

$a = {\begin{matrix} f_{1} (4 t), & 0 \leq t \leq 1 / 4 \\ f_{2} (4 t - 1), & 1 / 4 < t \leq 1 / 2 \\ f_{3} (2 t - 1), & 1 / 2 < t \leq 1 \end{matrix}$

and:

$b = {\begin{matrix} f_{1} (2 t), & 0 \leq t \leq 1 / 2 \\ f_{2} (4 t - 2), & 1 / 2 < t \leq 3 / 4 \\ f_{3} (4 t - 3), & 3 / 4 < t \leq 1 \end{matrix}$

If we denote the homotopy by $H$ , we want $H (t, 0) = a (t), H (t, 1) = b (t)$ and $H (0, s) = H (1, s) = x_{0}$ . This homotopy is explicitly given by:

$H (t, s) = {\begin{matrix} f_{1} (\frac{4 t}{1 + s}), & 0 \leq t \leq (1 + s) / 4 \\ f_{2} (4 t - 1 - s), & (1 + s) / 4 < t \leq (2 + s) / 4 \\ f_{3} (\frac{4 t - 2 - s}{2 - s}), & (2 + s) / 4 < t \leq 1 \end{matrix}$

Graphical version

Uniform version

This version is a little stronger than the other versions. Let $L = Ω (X, x_{0})$ be the loop space of $(X, x_{0})$ , i.e., the space of all loops in $X$ based at $x_{0}$ under the compact-open topology. Then, consider the following two maps:

$A : L \times L \times L \to L, A (f_{1}, f_{2}, f_{3}) = (f_{1} * f_{2}) * f_{3}$

and:

$B : L \times L \times L \to L, B (f_{1}, f_{2}, f_{3}) = f_{1} * (f_{2} * f_{3})$

Then, the maps $A$ and $B$ are homotopic maps. This is part of the proof that $Ω (X, x_{0})$ is a H-space, which is a homotopy variant of topological monoid.

@@ Line 8: / Line 8: @@
 <math>a</math> and <math>b</math> are homotopic loops, i.e., they are in the same homotopy class of loops based at <math>x_0</math>.
+This version is essentially the ''associativity'' part of showing that the [[fact about::fundamental group]] of a based topological space is indeed a group.
 ===Constructive/explicit version===
@@ Line 19: / Line 22: @@
 <math>b = \lbrace\begin{array}{rl} f_1(2t), & 0 \le t \le 1/2 \\ f_2(4t - 2), & 1/2 < t \le 3/4 \\ f_3(4t - 3), & 3/4 < t \le 1 \\\end{array}</math>
-If we denote the homotopy by <math>H</math>, we want <math>H(t,0) = a(t), H(t,1) = b(t)</math> and <math>H(0,s) = H(1,s) = x_0</math>. This homotopy is explicitly given by {{fillin}}
+If we denote the homotopy by <math>H</math>, we want <math>H(t,0) = a(t), H(t,1) = b(t)</math> and <math>H(0,s) = H(1,s) = x_0</math>. This homotopy is explicitly given by:
+<math>H(t,s) = \lbrace\begin{array}{rl} f_1\left(\frac{4t}{1 + s}\right), & 0 \le t \le (1 + s)/4 \\ f_2(4t - 1 - s), & (1 + s)/4 < t \le (2 + s)/4 \\ f_3\left(\frac{4t - 2 - s}{2 - s}\right), & (2 + s)/4 < t \le 1 \\\end{array}</math>
 ===Graphical version===
-{{fillin}}
+[[File:Associativityhomotopy.png|350px]]
+===Uniform version===
+This version is a little stronger than the other versions. Let <math>L = \Omega(X,x_0)</math> be the [[fact about::loop space of a based topological space|loop space]] of <math>(X,x_0)</math>, i.e., the space of all loops in <math>X</math> based at <math>x_0</math> under the [[compact-open topology]]. Then, consider the following two maps:
+<math>\! A:L \times L \times L \to L, \qquad A(f_1,f_2,f_3) = (f_1 * f_2) * f_3</math>
+and:
+<math>\! B:L \times L \times L \to L, \qquad B(f_1,f_2,f_3) = f_1 * (f_2 * f_3)</math>
+Then, the maps <math>A</math> and <math>B</math> are [[homotopic maps]]. This is part of the proof that <math>\Omega(X,x_0)</math> is a [[H-space]], which is a homotopy variant of [[topological monoid]].