Orbit spaces of Torus isomorphic to torus and the klein bottle Let $T=S^1\times S^1$  be the torus. 


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*Find a group G of homeomorphisms of $T$ such that $T/G$ is homeomorphic to $T$ and $|G|=2$.

*Find a group G of homeomorphisms of $T$ such that $T/G$ is homeomorphic to the Klein bottle $K$ and $|G|=2$.
For the first question I am considering $G=\{1,\phi\}$ where $\phi:S^1\times S^1$ is defined as 
$$\phi:(z_1,z_2)\mapsto (z_1,-z_2)$$
Is my answer correct?
I don't have any idea about the second question. Can you help?
 A: Your first answer looks right. If you think of the torus as $D = [0, 2\pi] \times [0, 2\pi]$ with edges identified in the usual way, then a fundamental domain for the first action looks like $[0, 2\pi] \times [0, \pi]$, sitting as the lower half of $D$. And when you apply the action of $\varphi$, this lower half gets translated upwards, so that if you'd drawn an arrow on the upper edge pointing right, it'd still point to the right.
And that "still points right" is why the quotient is still $T^2$.
Can you think of a rectangle (with directed labels on its edges) whose quotient is $K$? Can you make this rectangle the fundamental domain for a group acting on $T^2$ similarly to what you did before?
A: *

*If $a:\mathbb{R}^2 \to \mathbb{R}^2$ is given by $a: (x,y) \mapsto (\frac{1}{2}+x,\frac{1}{2} + y)$ then $a^2 = id$ mod $\mathbb{Z}^2$. The induced action $[a]:\mathbb{R}^2/\mathbb{Z}^2 \to \mathbb{R}^2/\mathbb{Z}^2$ is given by $[a]:(z,w)\mapsto (-z,-w)$ with $[a]^2 = id$.

*If $b:\mathbb{R}^2 \to \mathbb{R}^2$ is given by $a: (x,y) \mapsto (-x,\frac{1}{2} + y)$ then $b^2=id$ mod $\mathbb{Z}^2$. The induced action $[b]:\mathbb{R}^2/\mathbb{Z}^2 \to \mathbb{R}^2/\mathbb{Z}^2$ is given by $[b]:(z,w)\mapsto (\overline{z},-w)$ with $[b]^2 = id$. 

