Poincare homology sphere

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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 $SO(4,\mathbb {R} )$ 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 $SO(3,\mathbb {R} )$ (which can be identified with real projective three-dimensional space $\mathbb {R} \mathbb {P} ^{3}$) by the icosahedral group of order 60, with its natural embedding as a subgroup of $SO(3,\mathbb {R} )$. The icosahedral group is isomorphic to the alternating group $A_{5}$. (see A5 on Groupprops).