Poincare homology sphere

Definition
The Poincare homology sphere is a the unique three-dimensional homology sphere that is not homeomorphic to the 3-sphere. Explicitly, it is defined as follows:


 * 1) It is the quotient of the 3-sphere by the binary icosahedral group, a perfect group of order 120 embedded naturally in $$SO(4,\R)$$ via its quaternionic representation. The binary icosahedral group is isomorphic to $$SL(2,5)$$ (see SL(2,5) on Groupprops and also learn about its quaternionic representation).
 * 2) It is the quotient of $$SO(3,\R)$$ (which can be identified with real projective three-dimensional space $$\R\mathbb{P}^3$$) by the icosahedral group of order 60, with its natural embedding as a subgroup of $$SO(3,\R)$$. The icosahedral group is isomorphic to the alternating group $$A_5$$. (see A5 on Groupprops).