Visualizing cylinder as orbit space of torus.

Question

Consider the torus obtained from $[-1,1]^2$ by identifying it's opposite edges in the same direction.Then how do I obtain cylinder from it by a $C_2$ action on it? Our instructor said that if we reflect torus through $xz$-plane then we get a cylinder as an orbit space but I am unable yo visualize it properly. Could anyone tell me what's going on here?

Thanks in advance.

Answer 1

If you cut a donut in a plane parallel to the table it sits in you get a top and bottom half - this is a bagel. If instead you cut it down a plane perpendicular to the table (how you usually cut things on your table), you get a left and right half - each is topologically a cylinder. The reflection across this plane identifies the left and right half, so the quotient space is basically just one of them.