# Homotopy between composites associated in different ways

## Contents

## Statement

### Existential version

Suppose are loops based at a point in a topological space . We can consider two differently associated products of these three loops:

and are homotopic loops, i.e., they are in the same homotopy class of loops based at .

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 and :

and:

If we denote the homotopy by , we want and . This homotopy is explicitly given by:

### Graphical version

### Uniform version

This version is a little stronger than the other versions. Let be the loop space of , i.e., the space of all loops in based at under the compact-open topology. Then, consider the following two maps:

and:

Then, the maps and are homotopic maps. This is part of the proof that is a H-space, which is a homotopy variant of topological monoid.