# 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

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

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

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

$T := \mathbb{R}^n / \mathbb{Z}^n \,.$

For $n = 1$ this is the circle.

In this fashion each torus canonically carries the structure of an abelian group, in fact of an abelian Lie group

## Properties

Revised on November 7, 2013 10:44:11 by Urs Schreiber (188.200.54.65)