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,