# nLab torus

### Context

#### Topology

topology

algebraic topology

## Examples

#### Manifolds and cobordisms

manifolds and cobordisms

# Contents

## Definition

### In topology and differential geometry

The torus is the manifold (a smooth manifold, hence in particular also a topological 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. Notice that regarded as a group the torus carries a base point (the neutral element).

### In algebraic geometry

According to SGA3, for $X$ a base scheme then a 1-dimensional torus (in the sense of tori-as-groups) over it is a group scheme over $X$ which becomes isomorphic to the multiplicative group over $X$ after a faithfully flat group extension.

In (Lawson-Naumann 12, def. A.1) this is called “a form of” the multiplicative group over $X$.

By (Lawson-Naumann 12, prop. A.4) the moduli stack of 1-dimensional tori $\mathcal{M}_{1dtori}$ in this sense is equivalent to the delooping of the group of order two:

$\mathcal{M}_{1dtor} \simeq \mathbf{B}\mathbb{Z}/2\mathbb{Z} \,.$

The single nontrival automorphism of any 1-dimensional toris here is that induced by the canonical automorphism of the multiplicative group

$Aut(\mathbb{G}_m) \simeq \mathbb{Z}/2\mathbb{Z}$

which is the inversion involution (given by sending any element to its inverse element).

## References

The moduli stack of 1-dimensional tori in algebraic geometry is discussed (as the cusp point inside the moduli stack of elliptic curves) in

Revised on May 9, 2014 09:47:48 by Urs Schreiber (88.128.80.165)