Half rotation about $S^1$

Question

I am reading a proof.

A line in proof goes like this

" compose this map with a half rotation about $S^1$"

What does half rotation about circle mean? What is the explicit map?

Answer 1

Rotation about the unit circle can be thought of many ways. Probably the most common is to imagine the circle as a subset of the complex plane--in particular, all complex numbers whose modulus is 1. Then rotation about $S^1$ is just multiplication by $e^{i\theta}$, where $\theta$ is the angle of rotation.

Answer 2

Presumably this means "half of a full rotation around the circle" (in other words, a rotation by an angle of $\pi$ around the center of the circle). Explicitly, if $S^1=\{(x,y)\in\mathbb{R}^2:x^2+y^2=1\}$, then this is the map $f:S^1\to S^1$ defined by $f(x,y)=(-x,-y)$.