# $n$-torus

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,

 $(z_{1},z_{2},\ldots,z_{n})\cdot(w_{1},w_{2},\ldots,w_{n})=(z_{1}w_{1},z_{2}w_{% 2},\ldots,z_{n}w_{n})$
Title $n$-torus Ntorus 2013-03-22 13:59:58 2013-03-22 13:59:58 ack (3732) ack (3732) 7 ack (3732) Definition msc 22C05 msc 54B10 Torus