Homotopy between composites associated in different ways

From Topospaces
Jump to: navigation, search

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 = \lbrace\begin{array}{rl} f_1(4t), & 0 \le t \le 1/4 \\ f_2(4t - 1), & 1/4 < t \le 1/2 \\ f_3(2t - 1), & 1/2 < t \le 1 \\\end{array}

and:

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}

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) = \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}

Graphical version

Associativityhomotopy.png

Uniform version

This version is a little stronger than the other versions. Let L = \Omega(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, \qquad A(f_1,f_2,f_3) = (f_1 * f_2) * f_3

and:

\! B:L \times L \times L \to L, \qquad 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 \Omega(X,x_0) is a H-space, which is a homotopy variant of topological monoid.