The Circle

Abstract

So far, we have not actually computed any nontrivial fundamental groups. The purpose of this short chapter is to remedy this by computing the fundamental group of the circle. We will show, as promised, that $\pi _1 \left( {\mathbb{S}^1 ,1} \right)$ is an infinite cyclic group generated by the path class of the path ω that goes once around the circle counter-clockwise at constant speed. Thus each element of $\pi _1 \left( {\mathbb{S}^1 ,1} \right)$ is uniquely determined by an integer, called its “winding number,” which counts the net number of times and in which direction the path winds around the circle.