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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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I have updated my question with (hopefully) useful information –  Willem Noorduin Jul 29 '11 at 14:47
    
I can't give you a good reference off the top of my head, but the situation is relatively simple if you're considering actions of finite groups on manifolds. Either you have no fixed points, and then the quotient will be a manifold, or you have fixed points and then you get an orbifold (the latter is a gadget that "locally looks like a quotient of a manifold by a finite group"). Maybe you should have a look at Thurston's book on 3-manifolds. –  t.b. Jul 29 '11 at 14:54
    
For the geometry, you should probably have a look at Wolf, where such examples should be discussed. Maybe there are some computations of a more cohomological flavour in one of the three volumes of Greub-Halperin-Vanstone –  t.b. Jul 29 '11 at 15:14
    
I'd write it as either "pseudo-projective space", with a hyphen, or "pseudoprojective space". I.e. "pseudo-" is a prefix, not a stand-alone word. –  Michael Hardy Jul 30 '11 at 14:40
    
If I understand this correctly, if you identify every two antipodal points on the sphere, you get a projective space, but you want to identify sets of three points on the sphere if $m=3$, etc. But given a point on the sphere, how does one know which set of three it belongs to that should be identified? Just on the 2-sphere, for example? –  Michael Hardy Jul 30 '11 at 15:50
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