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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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You are correct about the sphere with four 'pinches'. However, an exactly correct representation of what the surface looks like can be made by simply folding the paper! If you consider the two sides of the fold as separate surfaces, and you flip from one side to the other as you draw your pencil over the boundary, then what you have is exactly the space you describe.

Topologically, this space is homeomorphic to the sphere. However, homeomorphisms do not capture information about the metric/distances. For this, we want an isometry from the quotient space that you describe to 3D space. This isometry is exactly the folded piece of paper I just described where you are allowed to treat the sides separately.

This structure is not technically a manifold because the corners (which would be called order-2 cones) are considered 'singularities'-- the metric is not well-defined in these places. The best way to describe what happens to a curve passing through one of these corners is to say it bounces back (this is what happens in the limit of a curve getting closer and closer to the corner-- try with a marker and see what I mean).

Instead, the type of object you are considering is called an 'orbifold' and is often (perhaps always; I'm not sure) obtainable by starting with a manifold (in this case the 2D Euclidean plane) and 'equating' or 'gluing' certain points together.

With no jargon spared, any manifold quotiented by a group which acts on it properly discontinuously will produce an orbifold. This particular orbifold you have unknowingly described is generated by three order 2 rotations $a,b,c$ lying on the corners of a right-angle isosceles triangle (up to some dilation in the $x$ and $y$ directions). If the group $G$ is generated by these rotations ($G = \langle \{a,b,c\}\rangle$) then the orbifold $O$ illustrated by your diagram is $\mathbb R^2 / G$ where $\mathbb R^2$ is a Riemannian manifold (I say Riemannian so that the notion of distance is preserved).

What's special about these 'singularities' is that they each have more than one stabilising element in $G$. All other points have only the identity in their stabiliser. Singularities (of cone types, and corner types) correspond precisely to those points with more than one stabiliser.

For more information on this topic, I suggest reading these Princeton notes by Thurston. Chapter 13 explains orbifolds.

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