It does not follow that X is simply connected, only that its fundamental group is perfect (see Hurewicz theorem).
Gluing each pair of opposite faces together using this identification yields a closed 3-manifold.
More intuitively, this means that the Poincaré homology sphere is the space of all geometrically distinguishable positions of an icosahedron (with fixed center and diameter) in Euclidean 3-space.
One can also pass instead to the universal cover of SO(3) which can be realized as the group of unit quaternions and is homeomorphic to the 3-sphere.
is the binary icosahedral group, the perfect double cover of I embedded in
The Poincaré homology sphere results from +1 surgery on the right-handed trefoil knot.
In 2003, lack of structure on the largest scales (above 60 degrees) in the cosmic microwave background as observed for one year by the WMAP spacecraft led to the suggestion, by Jean-Pierre Luminet of the Observatoire de Paris and colleagues, that the shape of the universe is a Poincaré sphere.
[3] Data analysis from the Planck spacecraft suggests that there is no observable non-trivial topology to the universe.
Ciprian Manolescu showed[5] that there is no such homology sphere with the given property, and therefore, there are 5-manifolds not homeomorphic to simplicial complexes.