Homology sphere

In algebraic topology, a homology sphere is an "n"-manifold "X" having the homology groups of an "n"-sphere, for some integer formula_1. That is,
and
Therefore "X" is a connected space, with one non-zero higher Betti number, namely, formula_4. 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. 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 formula_5). 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 formula_6 where formula_7 is the binary icosahedral group, the perfect double cover of I embedded in formula_8.
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.
As of 2016, the publication of data analysis from the Planck spacecraft suggests that there is no observable non-trivial topology to the universe.
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. the existence of such a homology 3-sphere was an unsolved problem. On March 11, 2013, Ciprian Manolescu posted a preprint on the ArXiv claiming to show 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 (see Galewski and Stern, A universal 5-manifold with respect to simplicial triangulations, in Geometric Topology (Proceedings Georgia Topology Conference, Athens Georgia, 1977, Academic Press, New York, pp 345–350)) is not triangulable.