nLab torus

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

$$T \coloneqq \mathbb{R}^2 / \mathbb{Z}^2$$

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

As a topological space this is the quotient topological space obtained from the square by identifying opposite sides:

graphics grabbed from Lawson 03

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

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

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).

As a homotopy type

As a homotopy type the torus is the product of two copies of the circle.

In homotopy type theory the torus can be formalized as the higher inductive type generated by a point base, two paths, $p$ and $q$, from base to itself, and an element $t$ of $p\cdot q = q \cdot p$. It has been formally shown (Sojakova15) that this type is equivalent to the product of two circles. For a treatment in cubical type theory, see (Licata-Brunierie).

$\,$

The stable homotopy type of the torus is the wedge sum of 2 circles and one 2-sphere (eg. Freed & Moore 13, Thm. 11.8):

(graphics from SS 22)

Properties

• The character group of a torus $T^n$ is isomorphic to its fundamental group, which is $\mathbb{Z}^n$. See also at Pontryagin duality and at moduli space of flat connections.

Proposition

One has

$$TC(T^n) =n+1$$

for the topological complexity.

(Farber 01, Theorem 13)

Related concepts

• solid torus

• complex torus

• elliptic curve

• toric variety

• noncommutative torus

References

• Terry Lawson, Topology: A Geometric Approach, Oxford University Press (2003) (pdf)

On the ordinary cohomology of topological tori:

• Qiaochu Yuan, The cohomology of the $n$-torus (2013)

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

• Tyler Lawson, Niko Naumann, Appendix A of Strictly commutative realizations of diagrams over the Steenrod algebra and topological modular forms at the prime 2 (arXiv:1203.1696)

Discussion of tori the homotopy type of the torus in homotopy type theory is in

• Kristina Sojakova, The equivalence of the torus and the product of two circles in homotopy type theory, (arXiv:1510.03918)

• Dan Licata, Guillaume Brunerie, A cubical approach to synthetic homotopy theory (pdf)

On 2-group-extensions of torus groups:

• Nora Ganter, Categorical Tori, SIGMA 14 (2018), 014, 18 (arXiv:1406.7046)

On the stable homotopy type of the torus:

• Daniel Freed, Gregory Moore, Thm. 11.8 of: Twisted equivariant matter, Ann. Henri Poincaré (2013) 14: 1927 (arXiv:1208.5055)

On topological complexity of tori (or more general products of spheres)

• Michael Farber, Topological complexity of motion planning (2001), arXiv:math/0111197;