In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n ≥ 1 {\displaystyle n\geq 1} . That is,
H 0 ( X , Z ) = H n ( X , Z ) = Z {\displaystyle H_{0}(X,\mathbb {Z} )=H_{n}(X,\mathbb {Z} )=\mathbb {Z} }
and
H i ( X , Z ) = { 0 } {\displaystyle H_{i}(X,\mathbb {Z} )=\{0\}} for all other i. Therefore X is a connected space, with one non-zero higher Betti number, namely, b n = 1 {\displaystyle 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 the distinct configurations of a regular icosahedron, centered at the origin, 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 S 3 / I ~ {\displaystyle S^{3}/{\widetilde {I}}} where I ~ {\displaystyle {\widetilde {I}}} is the binary icosahedral group, the perfect double cover of I embedded in S 3 {\displaystyle 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 Surgery on a knot in the 3-sphere S3 with framing +1 or −1 gives a homology sphere. More generally, surgery on a link gives a homology sphere whenever the matrix given by intersection numbers (off the diagonal) and framings (on the diagonal) has determinant +1 or −1. If p, q, and r are pairwise relatively prime positive integers then the link of the singularity xp + yq + zr = 0 (in other words, the intersection of a small 3-sphere around 0 with this complex surface) is a Brieskorn manifold that is a homology 3-sphere, called a Brieskorn 3-sphere Σ(p, q, r). It is homeomorphic to the standard 3-sphere if one of p, q, and r is 1, and Σ(2, 3, 5) is the Poincaré sphere. The connected sum of two oriented homology 3-spheres is a homology 3-sphere. A homology 3-sphere that cannot be written as a connected sum of two homology 3-spheres is called irreducible or prime, and every homology 3-sphere can be written as a connected sum of prime homology 3-spheres in an essentially unique way. (See Prime decomposition (3-manifold).) Suppose that a 1 , … , a r {\displaystyle a_{1},\ldots ,a_{r}} are integers all at least 2 such that any two are coprime. Then the Seifert fiber space
{ b , ( o 1 , 0 ) ; ( a 1 , b 1 ) , … , ( a r , b r ) } {\displaystyle \{b,(o_{1},0);(a_{1},b_{1}),\dots ,(a_{r},b_{r})\}\,}
over the sphere with exceptional fibers of degrees a1, ..., ar is a homology sphere, where the b's are chosen so that
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