Editing Homology sphere (section)

===Construction===
A simple construction of this space begins with a [[dodecahedron]]. Each face of the dodecahedron is identified with its opposite face, using the minimal clockwise twist to line up the faces. [[Quotient space (topology)|Gluing]] each pair of opposite faces together using this identification yields a closed 3-manifold. (See [[Seifert–Weber space]] for a similar construction, using more "twist", that results in a [[hyperbolic 3-manifold]].)

Alternatively, the Poincaré homology sphere can be constructed as the [[Quotient space (topology)|quotient space]] [[SO(3)]]/I where I is the [[Icosahedral symmetry|icosahedral group]] (i.e., the rotational [[symmetry group]] of the regular [[icosahedron]] and dodecahedron, isomorphic to the [[alternating group]] A<sub>5</sub>). 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 [[quaternion]]s and is [[homeomorphic]] to the 3-sphere. In this case, the Poincaré homology sphere is isomorphic to <math>S^3/\widetilde{I}</math> where <math>\widetilde{I}</math> is the [[binary icosahedral group]], the perfect [[Double covering group|double cover]] of I [[Embedding|embedded]] in <math>S^3</math>.

Another approach is by [[Dehn surgery]]. The Poincaré homology sphere results from +1 surgery on the right-handed [[trefoil knot]].