Fundamental group

Definition

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 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 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.

Proof that this gives a group structure

To prove that the multiplication defined above does give a group structure, we note that there is a homotopy between the identity map on $[0,1]$ and any increasing homeomorphism on it. Thus any reparametrization of a curve is homotopic to the original curve. This can be used to show that the composition operation defined above is associative on homotopy classes of loops.

The inverse of a path is the same path traversed in the opposite direction, and the identity element is the homotopy class of the trivial loop.

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