# nLab torus

### Context

#### Topology

topology

algebraic topology

## Examples

#### Manifolds and cobordisms

manifolds and cobordisms

# Contents

## Definition

The torus is the manifold obtained as the quotient

$T:={ℝ}^{2}/{ℤ}^{2}$T := \mathbb{R}^2 / \mathbb{Z}^2

of the Cartesian plane, regarded as an abelian group, by the subgroup of pairs of integers.

More generally, for $n\in ℕ$ any natural number, the $n$-torus is

$T:={ℝ}^{n}/{ℤ}^{n}\phantom{\rule{thinmathspace}{0ex}}.$T := \mathbb{R}^n / \mathbb{Z}^n \,.

For $n=1$ this is the circle.

## Properties

The torus naturally inherits the structure of a Lie group.

Revised on July 10, 2012 11:32:11 by Urs Schreiber (82.113.121.81)