Manifolds and cobordisms
In topology and differential geometry
The torus is the manifold (a smooth manifold, hence in particular also a topological manifold) obtained as the quotient
of the Cartesian plane, regarded as an abelian group, by the subgroup of pairs of integers.
More generally, for any natural number, the -torus is
For 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 a base scheme then a 1-dimensional torus (in the sense of tori-as-groups) over it is a group scheme over which becomes isomorphic to the multiplicative group over after a faithfully flat group extension.
In (Lawson-Naumann 12, def. A.1) this is called “a form of” the multiplicative group over .
By (Lawson-Naumann 12, prop. A.4) the moduli stack of 1-dimensional tori in this sense is equivalent to the delooping of the group of order two:
The single nontrival automorphism of any 1-dimensional toris here is that induced by the canonical automorphism of the multiplicative group
which is the inversion involution (given by sending any element to its inverse element).
The moduli stack of 1-dimensional tori in algebraic geometry is discussed (as the cusp point inside the moduli stack of elliptic curves) in