Understanding the smoothness of Poincaré dodecahedral space I've been trying to wrap my head around Poincaré Dodecahedral Space, aka the Poincaré homology sphere, which was suggested as a cosmological model after the release of WMAP cosmic microwave background data but contradicted by the improved Planck mission data. This is commonly described as gluing opposite faces of a dodecahedron, e.g. this figure from The Shape of Space by Jeffrey R. Weeks (reproduced here):

Since this is a 3-manifold, it must locally look like euclidean space. However, I don't understand how this is possible near the corners. It seems like mapping points near a corner to the opposite sides of the dodecahedron will result in points far apart. For instance, given the labels from this figure (Threlfall & Seifert, 1931, pg 66, uploaded here):

if we consider the center corner I, it is surrounded by $C_1^+$, $C_3^+$ and $C_4^+$. However, applying the glue operation we get the other three I corners, each of which is bounded by a different set of faces.
Does this imply that it's not just the opposing faces which are glued, but that there are actually 4 equivalent positions for any point in the dodecahedron? That seems wrong to me, since then the actual space would be something like three pentagonal pyramids merged and you'd get a singularity at the center. What am I overlooking here?
 A: Take the more familiar example of a torus: in identifying opposite sides of a square, the points on the edges have two equivalent positions in the polyhedral complex.
The mathematical machinery is obfuscated by the intuitive meaning of the word "identify." In formality, one is mapping equivalence classes of points on the surfaces in the diagram that are being identified to points on the "actual" manifold.
In this example, the collections of "four equivalent positions" are the points of the space.
To how the intuitive notion of continuity is preserved, the fact is it's a 3-manifold, not a 2-manifold—so "close" points in a two-dimensional visualization of the manifold need not map to "close" points on the manifold itself. Indeed, even a 3-dimensional visualization need not have this property; only a special class of visualizations do.
You may be better served asking this on the math site as your question is purely mathematical, and physicists often have deeply unsatisfying answers to questions about the nature of mathematical objects, as answering those isn't exactly their job description.
