In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n\ge 1. That is, and Therefore X is a connected space, with one non-zero higher Betti number, namely, b_n=1. 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 using homology with rational coefficients.

Poincaré homology sphere

The Poincaré homology sphere (also known as Poincaré dodecahedral space) is a particular example of a homology sphere, first constructed by Henri Poincaré. Being a spherical 3-manifold, it is the only homology 3-sphere (besides the 3-sphere itself) with a finite fundamental group. Its fundamental group is known as the binary icosahedral group and has order 120. Since the fundamental group of the 3-sphere is trivial, this shows that there exist 3-manifolds with the same homology groups as the 3-sphere that are not homeomorphic to it.

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. 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 SO(3)/I where I is the icosahedral group (i.e., the rotational symmetry group of the regular icosahedron and dodecahedron, isomorphic to the alternating group A5). 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 quaternions and is homeomorphic to the 3-sphere. In this case, the Poincaré homology sphere is isomorphic to where is the binary icosahedral group, the perfect double cover of I embedded in S^3. Another approach is by Dehn surgery. The Poincaré homology sphere results from +1 surgery on the right-handed trefoil knot.

Cosmology

In 2003, lack of structure on the largest scales (above 60 degrees) in the cosmic microwave background as observed for one year by the WMAP spacecraft led to the suggestion, by Jean-Pierre Luminet of the Observatoire de Paris and colleagues, that the shape of the universe is a Poincaré sphere. In 2008, astronomers found the best orientation on the sky for the model and confirmed some of the predictions of the model, using three years of observations by the WMAP spacecraft. Data analysis from the Planck spacecraft suggests that there is no observable non-trivial topology to the universe.

Constructions and examples

Invariants

Applications

If A is a homology 3-sphere not homeomorphic to the standard 3-sphere, then the suspension of A is an example of a 4-dimensional homology manifold that is not a topological manifold. The double suspension of A is homeomorphic to the standard 5-sphere, but its triangulation (induced by some triangulation of A) is not a PL manifold. In other words, this gives an example of a finite simplicial complex that is a topological manifold but not a PL manifold. (It is not a PL manifold because the link of a point is not always a 4-sphere.) Galewski and Stern showed that all compact topological manifolds (without boundary) of dimension at least 5 are homeomorphic to simplicial complexes if and only if there is a homology 3 sphere Σ with Rokhlin invariant 1 such that the connected sum Σ#Σ of Σ with itself bounds a smooth acyclic 4-manifold. Ciprian Manolescu showed that there is no such homology sphere with the given property, and therefore, there are 5-manifolds not homeomorphic to simplicial complexes. In particular, the example originally given by Galewski and Stern is not triangulable.

Selected reading