Fundamental group

Symbol-free definition
The fundamental group of a topological space with respect to a basepoint is defined as the group of homotopy classes of loops centered at that basepoint, where composition of two loops is their juxtaposition (doing one after the other).

Definition with symbols
Let $$X$$ be a topological space and $$x_0$$ a point in $$X$$. The fundamental group of $$X$$, denoted as $$\pi_1(X, x_0)$$ is defined as the group of homotopy classes of loops in $$X$$ starting and ending at $$x_o$$, where the product of two homotopy classes of loops is the homotopy class of a loop obtained by first completing the first loop, then completing the second loop.

As a functor
The map sending a given pointed space (topological space with a specified point in it) to its fundamental group is a functor from the category of pointed spaces to the category of groups.

Isomorphic fundamental groups
If two points are in the same path-component in the topological space, the fundamental groups with respect to those points are isomorphic. In fact, any path from one of these points to the others defines an isomorphism of their fundamental groups. Moreover, for a given point, any loop about that point defines an inner automorphism of the fundamental group with itself.

Correspondence with properly discontinuous group actions
The fundamental group of a path-connected, locally path-connected and semilocally simply connected topological space naturally acquires a properly discontinuous group action on the universal covering space of that space.

Given a simply connected topological space $$X$$, one is often interested in those spaces which have that as the universal covering space and a given fundamental group $$G$$. Finding all such spaces is equivalent to finding all properly discontinuous group actions of $$G$$ on $$X$$.

Fundamental group operator
The fundamental group operator takes as input a property of topological spaces and outputs the property of being a group that occurs as the fundamental group of a topological space with that property.

Simply connected space
A topological space is said to be simply connected if it is path-connected and its fundamental group is trivial. In other words, every loop in the topological space can be contracted to the trivial loop.