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)