chan-ho suh

You appear to be talking about Thurston's Princeton notes, but he also has
a book with a similar title of _Three-dimensional Geometry and Topology_.
The conceptual basics for what you're looking for is in both, but the
latter I think is more appropriate for what you want. It talks about
Poincare dodecahedral space, Seifert-Weber dodecahedral space, etc., right
from the beginning.

The Poincare dodecahedral space is obtained by gluing opposite faces of a
dodecahedron with the minimal clockwise turn needed to line up the
pentagons, whereas the Seifert-Weber space is obtained similarly by
additionally rotating with a 1/5 clockwise twist after having lined up the
faces as previously. The first corresponds to a tiling of the 3-sphere by
dodecahedrons and the second corresponsds to a tiling of hyperbolic space.
This is explained thoroughly in Thurston's book.

Since you mention you are interested in cosmology, perhaps Jeff Week's
book, _The Shape of Space_ would be a useful read.