Homology sphere

Encyclopedia : H : HO : HOM : Homology sphere

In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n ≥ 1. That is, we have

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

and

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

Therefore X is a connected space, with one non-zero higher Betti number: b_n. It does not follow that X is simply connected, only that its fundamental group is perfect (see Hurewicz theorem).

A rational homology sphere is defined similarly but we use homology with rational coefficients.

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.

A simple construction of this space, which makes clear the term "dodecahedral space", begins with a dodecahedron. Each face of the dodecahedron can be identified with its opposite face by using the minimal clockwise twist to line up the faces. Glue each pair of opposite faces together using this identification. After this gluing, the result is a closed 3-manifold.

The Poincaré sphere is a spherical 3-manifold. See Seifert-Weber space for a similar construction (using a different amount of "twist") 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.

From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
All text is available under the terms of the GNU Free Documentation License See Wikipedia Copyrights for details.