Fundamental group

Basic definition
The fundamental group of a based topological space $$(X,x_0)$$ is defined as follows:


 * As a set, it is the set of all homotopy classes of loops at $$x_0$$ in $$X$$. The homotopy class of a loop $$f$$ is denoted $$[f]$$. Note here that homotopy class of loop in particular means that at every stage of the homotopy, we must have a loop based at $$x_0$$. In particular, it is not the same thing as the intersection with loops based at $$x_0$$ of homotopy classes of paths in $$X$$.
 * The group structure is obtained as follows: the composite of two loops is obtained by first traversing the first loop, and then traversing the second loop. Explicitly, if $$f_1, f_2:[0,1] \to X$$ are the two loops, then the composite $$f_1 * f_2$$ of these is the loop given by $$t \mapsto f_1(2t)$$ for $$0 \le t \le 1/2$$ and $$t \mapsto f_2(2t - 1)$$ for $$1/2 \le t \le 1$$. Continuity of this new loop follows from the gluing lemma for closed subsets.

When the topological space is path-connected, the fundamental groups at any two basepoints are isomorphic.

Proof that this gives a group structure
All loops here are based at $$x_0$$.

One nice thing about all these homotopies is that they do not depend on additional properties of the ambient space, and the homotopies do not use any points of the space other than those used in the original loops.

Alternative definitions
This definition need not work for all topological spaces; however, it works for all compactly generated Hausdorff spaces and possibly many others.

The fundamental group of a based topological space $$(X,x_0)$$ is defined as the defining ingredient::space of path components of the loop space of $$(X,x_0)$$, i.e.:

$$\! \pi_1(X,x_0) = \pi_0(\Omega(X,x_0))$$

The group structure arises as the structure induced on the quotient by the natural multiplication structure on $$\Omega(X,x_0)$$. Because that is a H-space under the multiplication, $$\pi_0$$ of the space gets a monoid structure. It turns out that this monoid structure is also a group.

This definition can be reconciled with the usual definition as follows: paths in the loop space of a based topological space are the same thing as homotopies of based loops in the original space. Thus, the set of path components of the loop space based at a point is the same thing as the set of homotopy classes of loops based at the point. Further, the composition used to give a H-space structure to $$\Omega(X,x_0)$$ descends precisely to the group multiplication we use to define the fundamental group.

Topology on the fundamental group
The definition of the fundamental group as the space of path components of $$\Omega(X,x_0)$$ gives a topology on the fundamental group. It turns out, though, that if the path component of $$x_0$$ in $$X$$ is a locally path-connected space, then so is $$\Omega(X,x_0)$$, in which case the fundamental group has a discrete topology. Since the spaces we typically study (such as manifolds) are locally path-connected, the fundamental group is habitually viewed as a discrete group.

Omission of basepoint
For a path-connected space, the fundamental groups at two different points are isomorphic. Moreover, any path between these two points yields an isomorphism, and any two such isomorphisms have quotient equal to an inner automorphism of the fundamental group at one point. When viewing the fundamental group as an abstract group, it is thus often customary to omit the basepoint for a path-connected space.

More generally, for a topological space where all the path components are homeomorphic (or even more generally, where they are all homotopy-equivalent spaces), the isomorphism class of the fundamental group is independent of the choice of basepoint. In particular, this is true for homogeneous spaces such as topological groups.

For a path-connected space with abelian fundamental group, the fundamental groups at any two points are canonically identified and thus it makes even more sense to omit the basepoint.

Examples
Most of the examples of spaces below are topological spaces that are either path-connected or where all the path components are homeomorphic. Thus, the value of the fundamental group is independent of the choice of basepoint.

Related properties of topological spaces

 * A simply connected space is a topological space whose fundamental group is trivial
 * For a H-space and hence in particular for any space that arises as a loop space, and for any topological monoid, the fundamental group is Abelian

Aspects of the fundamental group

 * Fundamental group functor
 * Actions of the fundamental group