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| In [[algebraic topology]], a '''homology sphere''' is an ''n''-[[manifold]] ''X'' having the [[homology group]]s of an ''n''-[[sphere]], for some integer ''n'' ≥ 1. That is,
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| :''H''<sub>0</sub>(''X'','''Z''') = '''Z''' = ''H''<sub>''n''</sub>(''X'','''Z''')
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| :''H''<sub>''i''</sub>(''X'','''Z''') = {0} for all other ''i''.
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| Therefore ''X'' is a [[connected space]], with one non-zero higher [[Betti number]]: ''b<sub>n</sub>''. It does not follow that ''X'' is [[simply connected]], only that its [[fundamental group]] is [[perfect group|perfect]] (see [[Hurewicz theorem]]).
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| A '''rational homology sphere''' is defined similarly but using homology with rational coefficients.
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| ==Poincaré homology sphere==
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| <!-- Henri Poincaré links here -->
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| The [[Henri Poincaré|Poincaré]] homology sphere (also known as Poincaré dodecahedral space) is a particular example of a homology sphere. 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 the [[Poincaré conjecture]] cannot be stated in homology terms alone.
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| ===Construction===
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| 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. [[Quotient space|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]].)
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| Alternatively, the Poincaré homology sphere can be constructed as the [[quotient space]] [[SO(3)]]/I where I is the [[Icosahedral symmetry|icosahedral group]] (i.e. the rotational [[symmetry group]] of the regular [[icosahedron]] and dodecahedron, isomorphic to the [[alternating group]] ''A''<sub>5</sub>). 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 [[quaternion]]s and is [[homeomorphic]] to the 3-sphere. In this case, the Poincaré homology sphere is isomorphic to ''S''<sup>3</sup>/Ĩ where Ĩ is the binary icosahedral group, the perfect [[Double covering group|double cover]] of I [[Embedding|embedded]] in ''S''<sup>3</sup>.
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| Another approach is by [[Dehn surgery]]. The Poincaré homology sphere results from +1 surgery on the right-handed [[trefoil knot]].
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| ===Cosmology===
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| In 2003, lack of structure on the largest scales (above 60 degrees) in the [[cosmic microwave background]] as observed for one year by the [[Wilkinson Microwave Anisotropy Probe|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.<ref name="physwebLum03">[http://physicsworld.com/cws/article/news/18368 "Is the universe a dodecahedron?"], article at PhysicsWorld.</ref><ref name="Nat03">{{cite journal
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| | last = Luminet
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| | first = Jean-Pierre
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| | authorlink = Jean-Pierre Luminet
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| | coauthors = [[Jeffrey Weeks (mathematician)|Jeff Weeks]], Alain Riazuelo, Roland Lehoucq, Jean-Phillipe Uzan
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| | title = Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background
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| | journal = Nature
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| | volume = 425
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| | issue = 6958
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| | pages = 593–595
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| | publisher = [[Nature]]
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| | location =
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| | date = 2003-10-09
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| | arxiv = astro-ph/0310253
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| | issn =
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| | doi = 10.1038/nature01944
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| | id =
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| | pmid = 14534579
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| | bibcode=2003Natur.425..593L}}</ref> 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.<ref name="RBSG08">{{cite journal
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| | last =Roukema
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| | first =Boudewijn
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| | authorlink =
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| | coauthors = Zbigniew Buliński, Agnieszka Szaniewska, Nicolas E. Gaudin
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| | title =A test of the Poincare dodecahedral space topology hypothesis with the WMAP CMB data
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| | journal = Astronomy and Astrophysics
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| | volume =482
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| | issue =3
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| | pages =747–753
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| | publisher =
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| | year = 2008
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| | arxiv =0801.0006
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| | doi =10.1051/0004-6361:20078777
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| | id =
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| | bibcode=2008A&A...482..747L}}</ref>
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| However, there is no strong support for the correctness of the model, as yet.
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| ==Constructions and examples==
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| *Surgery on a knot in the 3-sphere ''S''<sup>3</sup> with framing +1 or − 1 gives a homology sphere.
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| *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.
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| *If ''p'', ''q'', and ''r'' are pairwise relatively prime positive integers then the link of the singularity ''x''<sup>''p''</sup> + ''y''<sup>''q''</sup> + ''z''<sup>''r''</sup> = 0 (in other words, the intersection of a small 5-sphere around 0 with this complex surface) is a homology 3-sphere, called a [[Egbert Brieskorn|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.
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| *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)]].)
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| *Suppose that ''a''<sub>1</sub>, ..., ''a''<sub>''r''</sub> are integers all at least 2 such that any two are coprime. Then the [[Seifert fiber space]]
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| :: <math>\{b, (o_1,0);(a_1,b_1),\dots,(a_r,b_r)\}\,</math>
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| :over the sphere with exceptional fibers of degrees ''a''<sub>1</sub>, ..., ''a''<sub>''r''</sub> is a homology sphere, where the ''b'''s are chosen so that
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| :: <math>b+b_1/a_1+\cdots+b_r/a_r=1/(a_1\cdots a_r).</math>
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| :(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 3-sphere; otherwise they are distinct non-trivial 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 3-sphere with infinite fundamental group that has a [[Thurston geometry]] modeled on the universal cover of [[SL2(R)|''SL''<sub>2</sub>('''R''')]].
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| ==Invariants==
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| *The [[Rokhlin invariant]] is a '''Z'''/2'''Z''' valued invariant of homology 3-spheres.
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| *The [[Casson invariant]] is an integer valued invariant of homology 3-spheres, whose reduction mod 2 is the Rokhlin invariant.
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| ==Applications==
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| If ''A'' is a homology 3-sphere not homeomorphic to the standard 3-sphere, then the [[suspension (topology)|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 (topology)|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 (geometry)|link]] of a point is not always a 4-sphere.)
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| 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. {{as of|2013}} 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
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| 5-manifolds not homeomorphic to simplicial complexes. In particular, the example originally given by Galewski and
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| 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.
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| ==References==
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| {{Reflist}}
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| ==Selected reading==
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| * Emmanuel Dror, ''Homology spheres'', Israel Journal of Mathematics 15 (1973), 115–129. {{MathSciNet|0328926}}
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| * David Galewski, Ronald Stern [http://links.jstor.org/sici?sici=0003-486X%28198001%292%3A111%3A1%3C1%3ACOSTOT%3E2.0.CO%3B2-N ''Classification of simplicial triangulations of topological manifolds''], [[Annals of Mathematics]] 111 (1980), no. 1, pp. 1–34.
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| * [[Robion Kirby]], Martin Scharlemann, ''Eight faces of the Poincaré homology 3-sphere''. Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), pp. 113–146, [[Academic Press]], New York-London, 1979.
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| * [[Michel Kervaire]], ''[http://links.jstor.org/sici?sici=0002-9947%28196910%29144%3C67%3ASHSATF%3E2.0.CO%3B2-G Smooth homology spheres and their fundamental groups]'', [[Transactions of the American Mathematical Society]] 144 (1969) 67–72. {{MathSciNet|0253347}}
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| * Nikolai Saveliev, ''Invariants of Homology 3-Spheres'', Encyclopaedia of Mathematical Sciences, vol 140. Low-Dimensional Topology, I. Springer-Verlag, Berlin, 2002. {{MathSciNet|1941324}} ISBN 3-540-43796-7
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| ==External links==
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| *[http://www.eg-models.de/models/Simplicial_Manifolds/2003.04.001/_preview.html A 16-Vertex Triangulation of the Poincaré Homology 3-Sphere and Non-PL Spheres with Few Vertices] by [[Anders Björner]] and [[Frank H. Lutz]]
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| *Lecture by [[David Gillman]] on [http://media.pims.math.ca/realvideo-ram/science/2002/cascade/gillman/gillman.ram The best picture of Poincare's homology sphere ]
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| *[http://physicsworld.com/cws/article/print/23009 A cosmic hall of mirrors] - physicsworld (26 Sep 2005)
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| [[Category:Topological spaces]]
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| [[Category:Homology theory]]
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| [[Category:3-manifolds]]
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| [[Category:Spheres]]
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