.
Homology sphere
In algebraic topology, a homology sphere is an nmanifold X having the homology groups of an nsphere, for some integer n ≥ 1. That is,
 H_{0}(X,Z) = Z = H_{n}(X,Z)
and
 H_{i}(X,Z) = {0} for all other i.
Therefore X is a connected space, with one nonzero 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 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. Being a spherical 3manifold, it is the only homology 3sphere (besides the 3sphere itself) with a 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.
ConstructionAnother approach is by Dehn surgery. The Poincaré homology sphere results from +1 surgery on the righthanded trefoil knot.
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 3manifold. (See Seifert–Weber space for a similar construction, using more "twist", that results in a hyperbolic 3manifold.)
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 A_{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 3space. 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 3sphere. In this case, the Poincaré homology sphere is isomorphic to S^{3}/Ĩ where Ĩ 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 righthanded 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 JeanPierre Luminet of the Observatoire de Paris and colleagues, that the shape of the universe is a Poincaré sphere.[1][2] 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.[3] However, there is no strong support for the correctness of the model, as yet.
Constructions and examples
 Surgery on a knot in the 3sphere S^{3} 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 x^{p} + y^{q} + z^{r} = 0 (in other words, the intersection of a small 5sphere around 0 with this complex surface) is a homology 3sphere, called a Brieskorn 3sphere Σ(p, q, r). It is homeomorphic to the standard 3sphere 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 3spheres is a homology 3sphere. A homology 3sphere that cannot be written as a connected sum of two homology 3spheres is called irreducible or prime, and every homology 3sphere can be written as a connected sum of prime homology 3spheres in an essentially unique way. (See Prime decomposition (3manifold).)
 Suppose that a_{1}, ..., a_{r} are integers all at least 2 such that any two are coprime. Then the Seifert fiber space

 \( \{b, (o_1,0);(a_1,b_1),\dots,(a_r,b_r)\}\ \)
 over the sphere with exceptional fibers of degrees a_{1}, ..., a_{r} is a homology sphere, where the b's are chosen so that

 \( b+b_1/a_1+\cdots+b_r/a_r=1/(a_1\cdots a_r). \)
 (There is always a way to choose the b′s, and the homology sphere does not depend (up to isomorphism) on the choice of b′s.) If r is at most 2 this is just the usual 3sphere; otherwise they are distinct nontrivial homology spheres. If the a′s are 2, 3, and 5 this gives the Poincaré sphere. If there are at least 3 a′s, not 2, 3, 5, then this is an acyclic homology 3sphere with infinite fundamental group that has a Thurston geometry modeled on the universal cover of SL_{2}(R).
Invariants
The Rokhlin invariant is a Z/2Z valued invariant of homology 3spheres.
The Casson invariant is an integer valued invariant of homology 3spheres, whose reduction mod 2 is the Rokhlin invariant.
Applications
If A is a homology 3sphere not homeomorphic to the standard 3sphere, then the suspension of A is an example of a 4dimensional homology manifold that is not a topological manifold. The double suspension of A is homeomorphic to the standard 5sphere, 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 4sphere.)
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 4manifold. As of 2013 the existence of such a homology 3sphere was an unsolved problem. On March 11, 2013, Ciprian Manolescu posted a preprint on the ArXiv[4] claiming to show that there is no such homology sphere with the given property, and therefore, there are 5manifolds not homeomorphic to simplicial complexes. In particular, the example originally given by Galewski and Stern (see Galewski and Stern, A universal 5manifold 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.
References
"Is the universe a dodecahedron?", article at PhysicsWorld.
Luminet, JeanPierre; Jeff Weeks, Alain Riazuelo, Roland Lehoucq, JeanPhillipe Uzan (20031009). "Dodecahedral space topology as an explanation for weak wideangle temperature correlations in the cosmic microwave background". Nature 425 (6958): 593–595. arXiv:astroph/0310253. Bibcode:2003Natur.425..593L. doi:10.1038/nature01944. PMID 14534579.
Roukema, Boudewijn; Zbigniew Buliński; Agnieszka Szaniewska; Nicolas E. Gaudin (2008). "A test of the Poincare dodecahedral space topology hypothesis with the WMAP CMB data". Astronomy and Astrophysics 482 (3): 747–753. arXiv:0801.0006. Bibcode:2008A&A...482..747L. doi:10.1051/00046361:20078777.
Manolescu, Ciprian. "Pin(2)equivariant SeibergWitten Floer homology and the Triangulation Conjecture". arXiv:1303.2354. To appear in Journal of the AMS.
Selected reading
Emmanuel Dror, Homology spheres, Israel Journal of Mathematics 15 (1973), 115–129. MR 0328926
David Galewski, Ronald Stern Classification of simplicial triangulations of topological manifolds, Annals of Mathematics 111 (1980), no. 1, pp. 1–34.
Robion Kirby, Martin Scharlemann, Eight faces of the Poincaré homology 3sphere. Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), pp. 113–146, Academic Press, New YorkLondon, 1979.
Michel Kervaire, Smooth homology spheres and their fundamental groups, Transactions of the American Mathematical Society 144 (1969) 67–72. MR 0253347
Nikolai Saveliev, Invariants of Homology 3Spheres, Encyclopaedia of Mathematical Sciences, vol 140. LowDimensional Topology, I. SpringerVerlag, Berlin, 2002. MR 1941324 ISBN 3540437967
External links
A 16Vertex Triangulation of the Poincaré Homology 3Sphere and NonPL Spheres with Few Vertices by Anders Björner and Frank H. Lutz
Lecture by David Gillman on The best picture of Poincare's homology sphere
A cosmic hall of mirrors  physicsworld (26 Sep 2005)
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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