# Contents

## Definition

The circle group $𝕋$ is equivalently (isomorphically)

• the quotient group $ℝ/ℤ$ of the additive group of real numbers by the additive group of integers, induced by the canonical embedding $ℤ↪ℝ$;

• the unitary group $\mathrm{U}\left(1\right)$;

• the special orthogonal group $\mathrm{SO}\left(2\right)$;

• the subgroup of the group ${ℂ}^{×}$ of units of the field of complex numbers given by those of any fixed positive modulus (standardly $1$).

## Properties

For general abstract properties usually the first characterization is the most important one. Notably it implies that the circle group fits into a short exact sequence

$0\to ℤ\to ℝ\to 𝕋\to 0\phantom{\rule{thinmathspace}{0ex}}.$0 \to \mathbb{Z} \to \mathbb{R} \to \mathbb{T} \to 0 \,.

(On the other hand, the last characterization is usually preferred when one wants to be concrete.)

A character of an abelian group $A$ is simply a homomorphism from $A$ to the circle group.

A principal bundle with structure group the circle group is a circle bundle. The canonically corresponding associated bundle under the standard representation of $U\left(1\right)↪ℂ$ is a complex line bundle.

Revised on February 20, 2013 03:09:08 by Urs Schreiber (80.81.16.253)