notation for a torus

Question

I am trying to search for the meaning of this notation but unfortunately it seems that wikipedia even doesn't have it. The book I am following uses the following notation for a torus: $\mathbb{T}^d = \mathbb{R}^d/\mathbb{Z}^d$. What does this mean? I 've never seen this notation before.

Answer 1

This means that the torus is the quotient of the Lie group $\mathbb{R}^n$ by the discrete subgroup $\mathbb{Z}^n$. It has natural structure of a smooth manifold. This comes from a general theorem that any quotient $G/H$ of a Lie group by a closed Lie subgroup is a manifold.

If you don't need the smooth manifold structure, then you can consider this quotient just as a quotient of abstract (abelian) groups.