# nLab circle

### Context

#### Topology

topology

algebraic topology

## Examples

#### Manifolds and cobordisms

manifolds and cobordisms

# Contents

## Idea

The circle is a fantastic thing with lots and lots of properties and extra structures. It is a:

and it is one of the basic building blocks for lots of areas of mathematics, including:

## Definition

We consider the circle first as a topological space, then as the homotopy type represented by that space.

### As a topological space

###### Definition

The two most common definitions of the circle are:

1. It is the subspace of $ℂ$ consisting of those numbers of length $1$:

$U\left(1\right)≔\left\{z\in ℂ:\mid z\mid =1\right\}$U(1) \coloneqq \{z \in \mathbb{C} : {|z|} = 1\}

(of course, $ℂ$ can be identified with ${ℝ}^{2}$ and an equivalent formulation of this definition given; also, to emphasise the reason for the notation $U\left(1\right)$, $ℂ$ can be replaced by ${ℂ}^{*}={\mathrm{GL}}_{1}\left(ℂ\right)$)

2. It is the quotient of $ℝ$ by the integers:

${S}^{1}≔ℝ/ℤ$S^1 \coloneqq \mathbb{R}/\mathbb{Z}

The standard equivalence of the two definitions is given by the map $ℝ\to ℂ$, $t↦{e}^{2\pi it}$.

### As a homotopy type

As a homotopy type the circle is for instance the homotopy pushout

${S}^{1}\simeq *\coprod _{*\coprod *}*\phantom{\rule{thinmathspace}{0ex}}.$S^1 \simeq * \coprod_{* \coprod * } * \,.

A formalization as a higher inductive type in homotopy type theory is given by

Axiom circle : Type.

Axiom base : circle.
Axiom loop : base ~~> base.

See (Shulman).

## Properties

The circle is a compact, connected topological space. It is a $1$-dimensional smooth manifold (indeed, it is the only $1$-dimensional compact, connected smooth manifold). It is not simply connected. Its first homotopy group is the integers

${\pi }_{1}\left({S}^{1}\right)\simeq ℤ\phantom{\rule{thinmathspace}{0ex}}.$\pi_1(S^1) \simeq \mathbb{Z} \,.

(A proof of this in homotopy type theory is in Shulman P1S1.)

But the higher homotopy groups ${\pi }_{n}\left({S}^{1}\right)\simeq *$, $n>1$ all vanish (and so is a homotopy 1-type). The circle is a model for the classifying space for the abelian group $ℤ$, the integers.

The product of the circle with itself is the torus

$T={S}^{1}×{S}^{1}\phantom{\rule{thinmathspace}{0ex}}.$T = S^1 \times S^1 \,.

Generally, the $n$-torus is $\left({S}^{1}{\right)}^{n}$.

## References

A formalization in homotopy type theory is in

The proof that ${\pi }_{1}\left({S}^{1}\right)\simeq ℤ$ in this context is in

Revised on October 10, 2013 11:21:11 by Toby Bartels (64.89.53.104)