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 .
Given a (based) continuous map (that is, is continuous and ), one may define a group homomorphism by sending each loop to . It is easy to see that sends homotopic loops to homotopic loops (indeed if is a homotopy from to , then is a homotopy from to ), and thus is a well-defined map. Also clearly preserves concatenations, so is indeed a homomorphism.
Furthermore, it is easy to see that if and are maps and , then: and if is the identity map on and is the identity map on , then Thus we may in fact regard as a (covariant) functor from the category of based topological spaces to the category of groups.
One can also show that the induced map depends only on the homotopy type of , that is if are (based) homotopic maps that . Indeed, for any loop , if is a based homotopy from to , then is a based homotopy from to , and thus in .
In order for the fundamental group to be a useful topological concept, any two spaces that are topologically "the same" must have the same fundamental group. Specifically, if and are homeomorphic then and are isomorphic.
We will in fact show that and are isomorphic if and satisfy the weaker notion of equivalence: homotopy equivalence.
Say that and are based homotopy equivalent () with homotopy equivalences and . (By definition, this means that and .) Now consider the induced maps and . From the previous section we get that: and Therefore is the inverse of , so in particular must be an isomorphism. Hence .
This gives us a very useful method for distinguishing topological spaces: if and are topological spaces whose fundamental groups are not not isomorphic then and cannot be homeomorphic (and in fact, they cannot be homotopy equivalent). For instance, one can show that and (where is the n-sphere), and hence a circle is not homeomorphic to a sphere.