Here $Bbb{R}^2/Bbb{Z}^2$ is the quotient space obatined from $Bbb{R}^2$ by identifying points of $Bbb{Z}^2$ i.e. $(x,y)sim (x’,y’)iff (x,y),(x’,y’)inBbb{Z}^2$.

$S^1times S^1:={(z,w)| z,win S^1}$

I define $f:Bbb{R}^2to {S^1}times S^1$ by $f(x,y)=(e^{2pi i x},e^{2pi i y})$. $f$ is continuous and onto.

As $f(n,m)=(1,1) forall (n,m)in Bbb{Z}^2$ i.e. $f$ agrees on $Bbb{Z}^2$. By the property of quotient space, $f$ induces a continuous map $tilde{f}:mathbb{R}^2/Bbb{Z}^2to S^1times S^1$ such that $tilde{f}((x,y))=f(x,y)$. This map is onto as well. But this map is not injective. I couldn’t move forward from here.

Although I have observed one thing- instead of **only** identifying the points of $Bbb{Z}^2$ if we identify the points as follows-

$$(x,y)sim (x’,y’)iff x-x’,y-y’inBbb{Z}^2$$

Then we would have $Bbb{R}^2/simapprox S^1times S^1$, the same $f$ will give rise to this homomorphism.

Can anyone help me to solve the problem? Thanks for help in advance.