Perhaps the simplest object of study in algebraic topology is the fundamental group.
Let be a based, topological space (that is, is a topological space, and is some point in ). Note that some authors will require to be path-connected. Now consider all possible "loops" on that start and end at , i.e. all continuous functions with . Call this collection (the loop space of ). Now define an equivalence relation on by saying that if there is a (based) homotopy between and (that is, if there is a continuous function with , , and ). Now let be the set of equivalence classes of under .
Now define a binary operation (called concatenation) on by One can check that if and then , and so induces a well-defined binary operation on .
One can now check that the operation makes into a group. The identity element is just the constant loop , and the inverse of a loop is just the loop traversed in the opposite direction (i.e. the loop ). We call the fundamental group of .
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.
Independence from base point
At this point, one might wonder how significant the choice of base point, , was. As it turns out, as long as is path-connected, the choice of base point is irrelevant to the final group .
Indeed, pick consider any other base point . As is path connected, we can find a path from to . Let be the reverse path from to . For any , define by One can now easily check that is in fact a well-defined map , and furthermore, that it is a homomorphism. Now we may similarly define the map by . One can now easily verify that is the inverse of . Thus is an isomorphism, so .
Therefore (up to isomorphism), the group is independent of the choice of . For this reason, we often just write for the fundamental of .