Isomorphism between $\mathbb{T}$ and $\mathbb{R}^n/\Gamma$? In the Analysis on Manifolds via the Laplacian page $51$, they define the torus as $\mathbb{T} = \mathbb{R}^n/\Gamma$. This quotient is unintuitive that it defines the torus. Is there exists a natural isomorphism between $\mathbb{T}$ and $\mathbb{R}^n/\Gamma$? Could anyone be able to explain to me rigourously why the quotient defines the torus $\mathbb{T}$?

The point is to understand an isomorphism between the circle $S^1$ and $\mathbb{R/Z}$ first.  If you think of $S^1$ as the unit circle in $\mathbb{C}$ then the isomorphism can be defined using the map $e^{2\pi i x}$ and applying the isomorphism theorem for groups.
