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I don't know if "pseudo projective space" is a general accepted term, but I once read a book on general topology where the term was used for $\mathbb{S}^n / (\mathbb{Z}/m\mathbb{Z})$ (where you get the normal $\mathbb{R}\mathbb{P}^n$ when you set $m=2$).

Does anyone know of a good reference as to the (differential) topology and geometry of such beasts?

I think this may be an example of orbifolds, but this isn't my field exactly. I guess you get a bunch of singularities when $m > 2$ with which you have to deal.

Edit:

Reading in Sieradski (Introduction to Topology and Homotopy), where this example was coming from. He is talking about the disc $\mathbb{D}^2/eq$, where $eq$ is the equivalence relation coming from $\phi: \mathbb{S}^1 \rightarrow \mathbb{S}^1:z \mapsto z e^{2\pi/m} $ (a rotation of $2\pi/m$ radians. Thus the identification map $q: \mathbb{D}^2 \rightarrow P_m$ wraps the boundary 1-sphere m-times around its image $q(\mathbb{S}^1)$. This $P_m$ is called a pseudo-projective plane of order m.

My question is to references for geometry and (differential) topology of $P_n$, and higher-dimensional analogues.

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    Good question, Sieradski doesn't explain this really further, I guess when you have a sphere $\mathbb{S}^1$, you have to glue the points $z e^{2\pi/3}$, $z e^{4 \pi/3}$, $z e^{6 \pi/3}$ together to form one point in the quotient space. Then consider the object that its bounding. But saying this, I am not really sure if something interesting is going to happen with the inside)2011-08-01

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