Torus

The n-torus

One can easily generalize the torus to arbitrary dimensions. An n-torus is defined as a product of n circles:

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:mathbb{T}^n = S^1 imes S^1 imes cdots imes S^1

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The torus discussed above is the 2-torus. The 1-torus is just the circle. The 3-torus is rather difficult to visualize. Just as for the 2-torus, the n-torus can be described as a quotient of Rn under integral shifts in any coordinate. That is, the n-torus is Rn modulo the action of the integer lattice Zn (with the action being taken as vector addition). Equivalently, the n-torus is obtained from the n-cube by gluing the opposite faces together.

Related Topics:
Action - Lattice - Cube

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An n-torus is an example of an n-dimensional compact manifold. It is also an example of a compact abelian Lie group. This follows from the fact that the unit circle is a compact abelian Lie group (when identified with the unit complex numbers with multiplication). Group multiplication on the torus is then defined by coordinate-wise multiplication.

Related Topics:
Compact - Manifold - Abelian - Lie group - Unit circle - Complex number

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Toroidal groups play an important part in the theory of compact Lie groups. This is due in part to the fact that in any compact Lie group one can always find a maximal torus; that is, a closed subgroup which is a torus of the largest possible dimension.

Related Topics:
Maximal torus - Subgroup

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The fundamental group of an n-torus is a free abelian group of rank n. The k-th homology group of an n-torus is a free abelian group of rank n choose k. It follows that the Euler characteristic of the n-torus is 0 for all n. The cohomology ring H•(Tn,Z) can be identified with the exterior algebra over the Z-module Zn whose generators are the duals of the n nontrivial cycles.

Related Topics:
Fundamental group - Free abelian group - Homology group - Choose - Euler characteristic - Cohomology ring - Exterior algebra - Module

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