Homology sphere
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In algebraic topology, a homology sphere is a topological space X having the homology groups of an n-sphere, for some integer n ≥ 1. That is, we have
- H0(X,Z) and Hn(X,Z) are infinite cyclic
and
- Hi(X,Z) = {0} for all other i.
Therefore X is a connected space, with one non-zero higher Betti number: bn. 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.
Here is a simple way to construct the space that gave rise to the term "dodecahedral space": first note that every face of a dodecahedron has an opposite face, so every face can be glued to its opposite by the minimal clockwise twist needed to line up the faces. After this gluing, the result is a closed 3-manifold.
The Poincare sphere is a spherical 3-manifold. See Seifert-Weber space for a similar construction that results in a hyperbolic 3-manifold.
Alternatively, the Poincaré sphere can be constructed as the quotient space SO(3)/I where I is the icosahedral group (i.e. the symmetry group of the regular icosahedron and dodecahedron; isomorphic to the alternating group A5); less technically, this means that the Poincare sphere is the space of all possible positions of an icosahedron. Alternatively, one can pass to the universal cover of SO(3) which can be realized as the group of unit quaternions 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.
Another approach is by Dehn surgery. The Poincaré sphere results from +1 surgery on the right handed trefoil knot.