The Fundamental Group of Circle, Radius 1
Consider the space
Letand let a be a loop inwith base pointwhich goes clockwise once around the circle. Let the set of loops inbe labeledFor eachand define
whereis the identity function.
Note thatif and only ifForthere issuch that
Thus there is an isomorphismbetween the quotient groupand whereifandloop the same number of times in the same sense, given by h([n,a]) =n.
The groupis generated by the element