SU(2)/Γ ≅ SO(3)/A5
the Poincaré homology sphere is the space of all geometrically distinguishable positions of an icosahedron with fixed center and size in ℝ3.
Since we can think of SU(2) as the unit sphere in ℂ2, we have
SU(2)/Γ ⊂ ℂ2/Γ
ℂ2/Γ is the space of all geometrically distinguishable positions of an icosahedron with fixed center and arbitrary (possibly vanishing) size in ℝ3.
The singularity happens when the icosahedron shrinks to zero size!