an essentially unique way as Seifert fiber spaces: the quotient manifold is a sphere and there are 3 exceptional fibers of orders 2, 3, and 4. Icosahedral case The fundamental group is a product of a cyclic group of order m coprime to 30 with the binary icosahedral group (order 120) which has the presentation When m is 1, the manifold is the Poincaré homology sphere. These manifolds are uniquely determined by their fundamental groups. They can all be represented in an essentially unique way as Seifert fiber spaces: the quotient manifold is a sphere and there are 3 exceptional fibers of orders 2, 3, and 5. References Peter Orlik, Seifert manifolds, Lecture Notes in Mathematics, vol. 291, Springer-Verlag (1972). William Jaco, Lectures on 3-manifold topology William Thurston, Three-dimensional geometry and topology. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, New Jersey, 1997. Geometric topology Riemannian geometry Group theory 3-manifolds