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. $(x,y)\sim (x',y')\iff (x,y),(x',y')\in\Bbb{Z}^2$.
$S^1\times S^1:=\{(z,w)|\ z,w\in S^1\}$
I define $f:\Bbb{R}^2\to {S^1}\times S^1$ by $f(x,y)=(e^{2\pi i x},e^{2\pi i y})$. $f$ 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 $\tilde{f}:\mathbb{R}^2/\Bbb{Z}^2\to S^1\times S^1$ 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-
$$(x,y)\sim (x',y')\iff (x-x',y-y')\in\Bbb{Z}^2$$
Then we would have $\Bbb{R}^2/\sim\approx S^1\times S^1$, the same $f$ will give rise to this homomorphism.
Can anyone help me to solve the problem? Thanks for help in advance.
Best Answer
What you are observing is an ambiguity in the "slash" notation $/$ for quotient spaces.
In the context of $\mathbb R^2 / \mathbb Z^2$, you should think of the group $\mathbb Z^2$ acting on the space $\mathbb R^2$, where $(m,n) \in \mathbb Z^2$ takes the point $(x,y) \in \mathbb R^2$ to the point $(x',y') = (x+m,y+n) \in \mathbb R^2$. This action defines a decomposition of $\mathbb R^2$ into orbits: the orbit of $(x,y) \in \mathbb R^2$ is just the set of all $(x',y') \in \mathbb R^2$ such that $x-x',y-y' \in \mathbb Z^2$. And the meaning of $\mathbb R^2 / \mathbb Z^2$ in this context is that it is the orbit space of the action, namely the quotient space with respect to the decomposition of $\mathbb R^2$ into orbits.
As your question indicates, the slash notation is also used for a different kind of quotient: namely when one is given a topological space $X$ and a subset $A \subset X$, one forms the quotient $X/A$ whose decomposition elements are $A$ itself and all singleton subsets $\{x\} \subset X$ for which $x \in X-A$.