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I got stuck while reading this:

Consider the torus sitting in $\mathbb{R}^3$ like a donut on a table.

Then you see that it is invariant by a rotation of $180^\circ$ around an horizontal axis.

The quotient by such involution is a sphere.

My question is why the orbit space is sphere?

I couldn't understand how to visualize it?

For reference I want to add this math stack question Is it possible to obtain a sphere from a quotient of a torus? - see the first answer

Thank you.

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Shivani Sengupta
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  • vague hint: try to draw a fundamental domain for the action and see how its boundary should be glued to itself – Albert Mar 01 '17 at 10:58
  • (I can give further hints if you want) – Albert Mar 01 '17 at 11:00
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    Instead of a donut, take an inflated tire. Where the rotaional axis passes through the torus, pinch it together (because top and bottom get identified by the rotation). Also cut off half of the torus at the pinching lines (because the halves are identified. What you get is more like a tortellini/croissant and readily deformed into a standard sphere. – Hagen von Eitzen Mar 01 '17 at 11:05
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    ok I'll elaborate how I see it (which should be more or less similar to what @HagenvonEitzen has in mind). call $D$ your horizontal axis passing through the torus and choose a plane $P$ containing $D$. $P$ cuts the torus in 2 pieces, and each a fundamental domain for the action. Pick one, call it $F$: it's a curved cylinder, and its boundary is 2 circles centered at points of $D$. each circle is glued to itself by a rotation of 180 degrees, which can be seen as pinching the circles. so you get a cylinder with pinched ends (looking like a pillow, if you don't see the tortellini...) which is – Albert Mar 01 '17 at 11:12
  • homeomorphic to a sphere – Albert Mar 01 '17 at 11:15
  • please check if my understanding is correct.if we assume x-y plane divide torus in two equal symmetric halves.then upper part of x-y plane is identified with lower half of that plane and the inner and outer circle of the torus there we have flip action about x axis.so what is left we have lower half with flip action on boundary circle.so we will get sphere@Glougloubarbaki – Shivani Sengupta Mar 01 '17 at 13:27

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