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