This is homework so no answers please.
Consider the quotient map $\pi:\mathbb{R}^{2}\to \mathbb{R}^{2}/\sim$ ,where $(x_{1}y_{1})\sim (x_{2},y_{2})$ iff $x_{1}-x_{2},y_{1}-y_{2}\in \mathbb{Z}$. I showed that $\pi|_{(a,b)\times (c,d)}$, where $b-a,d-c\in (0,1)$, is a coordinate chart for $\mathbb{R}^{2}/\sim$.
The question is to: Find diffeomorphism between $\mathbb{R}^{2}/\sim$ and $\mathbb{S}_{1}\times \mathbb{S}_{1}$.
Any mistakes or shorcuts:
The most natural map I can think of is $f(x,y):=(e^{i2\pi (x)},e^{i2\pi (y)})$ from $\mathbb{R}^{2}/\sim\mapsto \mathbb{S}_{1}\times\mathbb{S}_{1}$ .
I showed well-defined (for different elements in the equivalence class), bijectivity and smoothness. Now I want inverse smoothness.
My problem is that its' inverse will be $\pi\circ (arg(e^{i2\pi x}),arg(e^{i2\pi y}))$ ($\pi$ was included so that it will map into $\mathbb{R}^{2}/\sim$)
and so the $(arg(e^{i2\pi x}),arg(e^{i2\pi y}))$ cannot be defined on all of $\mathbb{S}_{1}\times \mathbb{S}_{1}$.