# What makes Torus Special

For the past couple of days I have been encountering the word "Torus" quite often.

• I would like to know what special properties does the Torus possess that it is studied very much in Mathematics.

A recent article on Toral Automorphisms was given out to students by one of our Professors, which i have posted here.

This is one example which illustrates what properties "Torus" possesses. I am looking for more exciting properties of the Torus which makes it ubiquitous in Mathematics. Another question :

• Are Torus and Doughnuts both same? Or is there any topological difference between them.
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What an informative collection of answers this question engendered! –  Joseph O'Rourke Feb 10 '11 at 2:33
Q: "Are Torus and Doughnuts both same?" A: No! Doughnuts taste better! –  Tpofofn Feb 11 '11 at 2:07
The posted link is no longer valid.m –  JoeTaxpayer Jul 3 '14 at 1:14

It is a basic fact that any complex, connected, compact Lie group is a torus of some dimension. So, topologically and analytically, abelian varieties (in particular, elliptic curves) are tori. The fact that an elliptic curve over $\mathbb{C}$ is a torus enables one to determine easily what the group of torsion points looks like.

The torus is an easy example of a space with a nontrivial cohomology ring. The reason is that it is a tensor product (by the Kunneth theorem) of the cohomology ring of $S^1$ with itself, and consequently becomes an exterior algebra (this works for the torus in several dimensions).

The torus in two dimensions is also a simple interesting example of an orientable surface of positive genus and, indeed, any orientable surface of positive genus can be obtained from tori.

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Every complex, compact, connected Lie group is abelian? –  Sean Tilson Feb 9 '11 at 23:34
@Sean: Dear Sean, an easy way to see this (at the very beginning of Mumford's Abelian Varieties) is the following. Let $G$ be such a group, and consider the action of $G$ (the adjoint action) on the tangent space $T_e(G)$. This is holomorphic, but a holomorphic map from a compact connected complex manifold into affine space is constant. Thus the adjoint representation on $T_e(G)$ is trivial. From this the abelianness follows because one may see that if $\mathrm{Exp}(tv)|_{t \in \mathbb{C}}$ is a geodesic at the origin, then it is mapped under the Lie-homomorphism... –  Akhil Mathew Feb 10 '11 at 0:02
...$x \to yxy^{-1}$ to the geodesic $\mathrm{Exp}(tw)$ where $w$ is the adjoint of $y$ applied to the tangent vector $v$ (by general facts about morphisms of Lie groups), and consequently everything in the image of the exponential map is in the center. But the exponential map has image containing a neighborhood of the identity. From this it follows that the center is the entire group. –  Akhil Mathew Feb 10 '11 at 0:04
thanks, that helps a lot. I did not realize how strong a condition compactness of a complex manifold was. My Lie-fu is weak. –  Sean Tilson Feb 10 '11 at 3:20

(Below I only consider compact orientable surfaces, i.e., those than embed in $3$-space)

The torus is the only surface which can be endowed with a metric of vanishing curvature.

It is the only parallelizable surface.

It is the only surface which can be turned into a topological group.

(Is it the only surface where a differential equation can have all its orbits dense? one orbit dense?)

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Where "only" is taken over the compact surfaces, else you also have the cylinder and the plane. –  Willie Wong Feb 9 '11 at 19:19
I really had in mind throughtout only compact orientable surfaces---I'll edit that into the answer. –  Mariano Suárez-Alvarez Feb 9 '11 at 20:07

Tori (plural of torus) are mathematical objects which are very rich of structure because their algebraic and geometric features interact deeply and nicely: 1. Geometrically, they can be described has a product of copies of $S^1$, making evident their nature of compact variety, whose homology and cohomology are non-trivial,well-understood and "easy" to manipulate. 2. Algebraically, they're a group (a Lie group, in fact) and there are more or less obvious links with the multiplicative algebraic group and the theory of lattices. 3. Analitically, tori are the natural "home" of periodic functions, elliptic functions and modular forms. 4. Finally, under some restrictions, tori are also endowed with an arithmetic structure which is deeply intertwined with all the previous ones.

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One of your questions was "Is a torus a doughnut?" The 2-dimensional surface that we know as a torus is the 2-dimensional surface of a standard doughnut or bagel. It is also the 2-dimensional surface of a coffee cup up to homeomorphism (read topological equivalence). This is why we say that a topologist cannot tell the difference between a doughnut and a coffee cup.

If a piece of rope (or better extension cord) is tied loosely into a knot and the ends joined together (or plugged into one another, respectively) then the boundary of the resulting figure is represented as torus. Three dimensional compact spaces that have no incompressible tori are important because of the geometrization conjecture.

To paraphrase Homer (Simpson, not the poet), "MMM. Tori..."

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