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torus

Context

Topology

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:= 2/ 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 nn \in \mathbb{N} any natural number, the nn-torus is

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

For n=1n = 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 XX a base scheme then a 1-dimensional torus (in the sense of tori-as-groups) over it is a group scheme over XX which becomes isomorphic to the multiplicative group over XX after a faithfully flat group extension.

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

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

1dtorB/2. \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(𝔾 m)/2 Aut(\mathbb{G}_m) \simeq \mathbb{Z}/2\mathbb{Z}

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

Properties

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 April 9, 2014 07:54:58 by Urs Schreiber (145.116.131.80)