Source: Topology (2nd Economy Edition)
If we pick a point of the space X to serve as a “base point” and consider only those paths that begin and end at , the set of these path-homotopy classes does form a group under *. It will be called the fundamental group of X.
The important thing about the fundamental group is that it is a topological invariant of the space X, and it will be crucial in studying homeomorphism problems.
Definition of fundamental group:
Let X be a space; let be a point of X. A path in X that begins and ends at is called a loop based at . The set of path homotopy classes of loops based at , with operation *, is defined as the fundamental group of X relative to the base point . It is denoted by .
This group is called the first homotopy group of X. There is also a second homotopy group, and even groups for all .
An example of a fundamental group:
is the trivial group (the group consisting of just the identity). This is because if f is a loop in based at , the straight line homotopy is a path homotopy between f and the constant path at .
An interesting question (discussed in the next upcoming blog posts) would be how the group depends on the base point .