Perhaps the simplest object of study in algebraic topology is the fundamental group. Let be a path-connected topological space, and let be any point. Now consider all possible "loops" on that start and end at , i.e. all continuous functions with . Call this collection . Now define an equivalence relation on by saying that if there is a continuous function with , , and . We call a homotopy. Now define . That is, we equate any two elements of which are equivalent under .
Unsurprisingly, the fundamental group is a group. The identity is the equivalence class containing the map given by for all . The inverse of a map is the map given by . We can compose maps as follows: One can check that this is indeed well-defined.
Note that the fundamental group is not in general abelian. For example, the fundamental group of a figure eight is the free group on two generators, which is not abelian. However, the fundamental group of a circle is , which is abelian.
More generally, if is an h-space, then is abelian, for there is a second multiplication on given by , which is "compatible" with the concatenation in the following respect:
We say that two binary operations on a set are compatible if, for every ,
If share the same unit (such that ) then and both are abelian.