Fundamental group: Difference between revisions

Revision as of 22:50, 2 November 2007

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 $t \mapsto f_{1} (2 t)$ for $0 \leq t \leq 1 / 2$ and $t \mapsto f_{2} (2 t - 1)$ for $1 / 2 \leq t \leq 1$ . 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

@@ Line 6: / Line 6: @@
 * As a set, it is the set of all homotopy classes of [[loop]]s at <math>x_0</math> in <math>X</math>
-* The group structure is obtained as follows: the composite of two
+* 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 <math>f_1, f_2:[0,1] \to X</math> are the two loops, then the composite of these is the loop given by <math>t \mapsto f_1(2t)</math> for <math>0 \le t \le 1/2</math> and <math>t \mapsto f_2(2t - 1)</math> for <math>1/2 \le t \le 1</math>. 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 <math>[0,1]</math> 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 group|Abelian]]