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I understand the construction of a torus from a square by pasting opposite edges of a square and also its CW structure of that.

It's easy to imagine.

But how to understand the construction of a surface of genus 2g from a polygon with 4g sides and its CW structure. Because in the torus case by folding and pasting the square we get the object. This we can imagine.

But in the 2g genus case how to imagine the procedure.

Please help me to understand. Thanks in advance.

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    Maybe the best way of convince yourself is to drawn the $2g$ closed curves on the surface of genus $g$ which generates the homology group and remove it you should get something homeomorphic to a disk. From this it is not hard to imagine the reverse procedure for get a surface of genus $g$ from a disk (or a polygon).2017-03-23

2 Answers 2