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$n$-torus (Definition)

The $n$ -torus, denoted $T^n$ , is a smooth orientable $n$ dimensional manifold which is the product of $n$ 1-spheres, i.e. $T^n = \underbrace{ S^{1} \times \cdots \times S^{1}}_{n}$ .

Equivalently, the $n$ -torus can be considered to be $\mathbb{R}^n$ modulo the action (vector addition) of the integer lattice $\mathbb{Z}^n$ .

The $n$ -torus is in addition a topological group. If we think of $S^{1}$ as the unit circle in $\mathbb{C}$ and $T^n = \underbrace{ S^{1} \times \cdots \times S^{1}}_{n}$ , then $S^{1}$ is a topological group and so is $T^n$ by coordinate-wise multiplication. That is,

$\displaystyle (z_1,z_2,\ldots,z_n) \cdot (w_1,w_2,\ldots,w_n) = (z_1w_1,z_2w_2,\ldots,z_nw_n) $




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"$n$-torus" is owned by ack. [ full author list (4) | owner history (1) ]
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See Also: torus

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Cross-references: multiplication, unit circle, topological group, addition, lattice, integer, vector addition, action, product, manifold, orientable, smooth
There is 1 reference to this entry.

This is version 4 of $n$-torus, born on 2003-10-15, modified 2008-02-05.
Object id is 4820, canonical name is NTorus.
Accessed 2530 times total.

Classification:
AMS MSC22C05 (Topological groups, Lie groups :: Compact groups)
 54B10 (General topology :: Basic constructions :: Product spaces)

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