---
tag: 6SQXPE0JPN
kind: statement
role: theorem
name: Fundamental group of the circle
msc: [55Q52, 57M05]
uses: [S1XK3HXHFX, 060W5CBZP6, 0C4JS9RF5M, 08MG7H8JW5]
about: [S1XK3HXHFX, 060W5CBZP6]
sources:
  - url: https://en.wikipedia.org/wiki/Fundamental_group
---

## Statement

Let $S^1 = \set{z \in \C : \abs{z} = 1}$ be the unit [[circle|060W5CBZP6]], with base point $1$.
For $n \in \Z$, let $\omega_n \colon [0, 1] \to S^1$ be the loop $\omega_n(s) = e^{2\pi i n s}$.
Then the map $\Z \to \pi_1(S^1, 1)$, $n \mapsto [\omega_n]$, is a [[group isomorphism|08MG7H8JW5]] from the additive group of [[integers|0C4JS9RF5M]] to the [[fundamental group|S1XK3HXHFX]] of $S^1$.
In particular $\pi_1(S^1, 1) \cong \Z$, generated by the class of the loop that goes once around the circle.

## Remarks

The proof uses the [[covering map|3XZRRE165N]] $p \colon \R \to S^1$, $p(t) = e^{2\pi i t}$.
By the [[path lifting property|HSWWF35BEX]], a loop $\gamma$ at $1$ lifts to a unique path $\tilde\gamma$ in $\R$ with $\tilde\gamma(0) = 0$, and $\tilde\gamma(1)$ is an integer, the degree of $\gamma$.
Homotopy lifting shows that path homotopic loops have the same degree, so the inverse map $[\gamma] \mapsto \tilde\gamma(1)$ is well defined.
The lift of $\omega_n$ is $s \mapsto ns$.
The lift of a concatenation is obtained by translating and joining lifts, which gives the homomorphism property.
The degree is the topological form of the [[winding number|0NQSV2EWDR]].
Consequences include the Brouwer fixed point theorem in dimension $2$ and the fundamental theorem of algebra.

## Explanations

### L1 human

Every loop on a circle that starts and ends at a fixed point winds around the circle some whole number of times, counting clockwise turns as negative.
Two such loops can be deformed into each other exactly when they wind the same number of times.
Going around $m$ times and then $n$ times is going around $m + n$ times, so the loops behave like the integers under addition.

### L2 human

This is the first nontrivial computation of a fundamental group, and the model for computations through covering spaces.
It shows that the circle is not contractible and that it is not a retract of the closed disk.
With the [[Seifert–van Kampen theorem|S17DY98VF2]] it gives the fundamental groups of wedges of circles, of surfaces and of many CW complexes.
It is also the case $n = 1$ of the computation $\pi_n(S^n) \cong \Z$.