How to show $\mathbb R^n/\mathbb Z^n$ is diffeomorphic to torus $\mathbb T^n$?

Question

Suppose the additive group $\mathbb Z^n$ acts on $\mathbb R^n$ through translation. How to show $\mathbb R^n/\mathbb Z^n$ is diffeomorphic to torus $\mathbb T^n$? The translation action is given by $$\psi_g:\mathbb R^n\rightarrow \mathbb R^n,\ x\mapsto g+x.$$

Answer 1

Hint: From group theory point of view, the circle group denoted by $T$ and defined as $$\{z\in\mathbb C\mid|z|=1\}=\{\text{e}^{i\theta}\mid\theta\in\mathbb R\}$$ We can define a homomorphism $f$ as follows: $$f:\mathbb R\to T,~~~f(x)=\text{e}^{2\pi x i}$$ Show that $f$ is a surjection with kernel $\mathbb Z$.

Answer 2

Another perspective: $\mathbb{R}^n/\mathbb{Z}^n$ is a connected, compact, abelian Lie group of dimension $n$. How many of those do you know of?