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H₀(X,Z) and H_n(X,Z) are infinite cyclic

and

H_i(X,Z) = {0} for all other i.

Therefore X is a connected space, with one non-zero higher Betti number: b_n. It doesn't follow that X is simply connected, only that its fundamental group is perfect.

Poincaré sphere

The Poincaré sphere, or Poincaré dodecahedral space, is a particular example of a homology sphere. It is the only known homology 3-sphere (besides the 3-sphere itself) with finite fundamental group. Its fundamental group is known as the binary icosahedral group and has order 120. This shows the Poincaré conjecture cannot be stated in homology terms alone.

The Poincaré sphere can be constructed as the quotient space SO(3)/I where I is the icosahedral group (i.e. the symmetry groupThe symmetry group of a geometric figure is the group of congruencies under which it is invariant, with composition as the operation. The article on group theory also contains an explanation of the concept. In Euclidean geometry, discrete symmetry groups of the regular icosahedronAn icosahedron [aiks'hidrn] noun (plural: -drons, -dra [-dr]) is a polyhedron having 20 faces. The faces of a regular icosahedron are equilateral triangles. Etymology 16th Century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], "icosa'hedra and dodecahedronA dodecahedron is a Platonic solid composed of twelve pentagonal faces, with three meeting at each vertex. It has twenty vertices and thirty edges. Its dual polyhedron is the icosahedron. Canonical coordinates for the vertices of a dodecahedron centered a; isomorphic to the alternating groupIn mathematics an alternating group is the group of even permutations of a finite set. The alternating group on the set {1,. n is called the alternating group of degree n or the alternating group on n letters and denoted by A. For instance: {1234, 1342, 1 A₅). Alternatively, one can pass to the universal cover of SO(3) which can be realized as the group of unit quaternionQuaternion is also a musical composition by Sofia Gubaidulina In mathematics, the quaternions are a non-commutative extension of the complex numbers. They were first described by William Rowan Hamilton of Ireland in 1843. At first, the quaternions were res and is homeomorphic to the 3-sphere. In this case, the Poincaré sphere is isomorphic to S³/Ĩ where Ĩ is the binary isosahedral group, the perfect double cover of I living in S³.

One can also construct the Poincaré sphere by glueing together opposite faces of a solid dodecahedronA dodecahedron is a Platonic solid composed of twelve pentagonal faces, with three meeting at each vertex. It has twenty vertices and thirty edges. Its dual polyhedron is the icosahedron. Canonical coordinates for the vertices of a dodecahedron centered a via parallel translations and rotation by 36° clockwise.