# The Covering Transformation Theorem - Fundamental Group of the Circle

Recall from The Covering Transformation Theorem for Finding Fundamental Groups page that if $X$ is a topological space and $(\tilde{X}, p)$ is a universal cover of $X$ then:

(1)(Where $A(\tilde{X})$ denotes the set of covering transformations of $\tilde{X}$.)

We will now look at applying this result to find the fundamental group of the circle, $S^1$.

Recall that if $X = S^1$ then $\tilde{X} = \mathbb{R}$ is the universal cover of $X$ with the covering map $p : \tilde{X} \to X$ defined by:

(2)In a sense, we wrap $\mathbb{R}$ around on itself a countable number of times. For each $n \in \mathbb{Z}$ let $f_n : \mathbb{R} \to \mathbb{R}$ be defined for all $x \in \mathbb{R}$ by:

(3)Observe that for each $n \in \mathbb{N}$, $f_n : \mathbb{R} \to \mathbb{R}$ is a covering transformation of $\mathbb{R}$. Also note that $A(\tilde{X})$ is generated by $f_1$, and so:

(4)So we have finally proven that the fundamental group of the circle is isomorphic to $\mathbb{Z}$.