$n$-torus

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

Equivalently, the $n$-torus can be considered to be ${ℝ}^n$ 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 $ℂ$ and $T^n=\underset{n}{\underbrace{S^1\times \mathrm{\cdots }\times S^1}}$, then $S^1$ is a topological group and so is $T^n$ by coordinate-wise multiplication. That is,

$$(z_1,z_2,\mathrm{\dots },z_n)\cdot (w_1,w_2,\mathrm{\dots },w_n)=(z_1{w}_1,z_2{w}_2,\mathrm{\dots },z_n{w}_n)$$