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I'm gluing the edges of a square together with the caveat that there's a "fold" down the middle. I think this produces sort of a sphere with four "pinches". I'm wondering if my intuition is correct and if someone could provide a more rigorous foundation for what I'm trying to do and elaborate on the nature of the singularities.

Gluing diagram.

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I assume you are talking about orbifolds or at least surfaces with singularities, if you are talking about topological spaces then that is just a sphere.

On the first case, imagine glueing first $A$ sides. What do we have? It is a cylinder with one ''fold'', like a straight line along it. Now, when you glue $B$ and $C$ sides $\textit{two}$ new pinches appear on the intersection of C,A sides and B, A sides but the fold is still there. You can see those two pinches are cone points.

So it would be like a sphere with two pinches and a segment, or as if you take a pillow with its four pinches and ''join'' two of them by pressing on the pillow and getting the closed segment between them.

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