How to show that the Möbius group (or the general linear projective space $PGL(2,\mathbb{C})$ is not simply connected?
For a definition see: Möbius transformations. For more background see: projective linear group.
According to [Applied Mathematical Sciences 61] D. H. Sattinger, O. L. Weaver (auth.) - Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics (1986, Springer-Verlag) there is a closed path that cannot be shrunk to a point.
The path is given by the Möbius transformations $z \mapsto z e^{2 \theta}$ for $\theta \in [0, \pi] $, I think. But how do I show that it cannot be shrunk to a point in (in the Möbius group)?