torus and square How to identify Torus as the quotient of the unit square. Is there any explicit homeomorphism between the some quotient topology of the unit square and Torus. What is the quotient topology on the unit square in this case.
By torus I mean $ S^1 \times S^1$.
 A: Let $[0,1]^2$ be the unit square and let $\sim$ be the smallest equivalence relation such that $(0,t)\sim(1,t)$ and $(t,0)\sim(t,1)$ for all $t\in[0,1]$. Now $(S^1)^2\cong[0,1]^2/\sim$ and an explicit homeomorphism is given by $\phi\colon [0,1]^2/\sim\to (S^1)^2$ and $\psi\colon (S^1)^2\to[0,1]^2/\sim$, where
$$\phi\colon(\alpha,\beta) \mapsto ((\sin(2\pi \alpha),\cos(2\pi \alpha)),(\sin(2\pi \beta),\cos(2\pi \beta)))$$
which is indeed well defined since $\phi(0,t) = ((0,1),(\sin(2\pi t),\cos(2\pi t))) = \phi(1,t)$ and similarly $\phi(t,0)=\phi(t,1)$.
A: It is possible to find a torus as a quotient of a unit square. To see how to do this, take a sheet of paper, roll it so that the top side and lower side coincide and then roll this around.
To make this formal, we take the unit square $I^2$ and the equivalence relation $\sim$ given by $(x,0)\sim(x,1)$ to identify the lower and top edges of the square and by $(0,x)\sim(1,x)$ to identify the left with the right side. The quotient $I^2/\sim$ is then homeomorphic to a torus $S^1\times S^1$.
If you want to explicitly give a homeomorphism, you can use complex $e$-powers and describe $S^1\times S^1$ as $\{(e^{2\pi i t},e^{2\pi i s})\mid s,t\in I\}$. Using this an explicit homeomorphism can be given as $$\phi:I^2\to S^1\times S^1,\quad (t,s)\mapsto(e^{2\pi i t},e^{2\pi i s}).$$ When checking this is a homeomorphism, just be careful when taking care of the edges of $I^2$ that are glued together, in the same way that $e^{2\pi i}=e^0$.
