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I don't understand geometrically why the identification below let us generate the shape on the right can someone explain or give me some intuition ?enter image description here

  • Try gluing the $a$ pieces together, and then the $b$ pieces in your mind/with some paper or cloth. This should give you a handle of sorts – Jack Davies Jan 31 '16 at 09:54
  • Then you can maybe see that each pair of edges $a, b$ and $c,d$ and etc. gives you a handle. – Jack Davies Jan 31 '16 at 10:05

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Start with the double torus case. We have the following octagon:

enter image description here

Start by cutting it along the diagonal depicted in the picture by a red dotted line. You'll get two squares with an "edge", i.e., the red line. Once one identifies both of those pieces together one gets two punctured torii: that is, $\Bbb T^2$ with a small disk removed from each. The boundary of the removed disk is precisely the image of the red dotted line under the quotient map.

Thus, glue back two copies of $\Bbb T^2 - D^2$ along the red circle. This is the image of the octagon under the quotient map. It's thus connected sum of two torii - which is precisely the double torus.

Balarka Sen
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