Editing Homology sphere (section)

==Poincaré homology sphere==
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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. [[Quotient space (topology)|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 (topology)|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 <math>S^3/\widetilde{I}</math> where <math>\widetilde{I}</math> is the [[binary icosahedral group]], the perfect [[Double covering group|double cover]] of I [[Embedding|embedded]] in <math>S^3</math>.

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 [[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">[https://physicsworld.com/a/is-the-universe-a-dodecahedron/ "Is the universe a dodecahedron?"], article at PhysicsWorld.</ref><ref name="Nat03">{{cite journal
  | last1 = Luminet
  | first1 = Jean-Pierre
  | author1-link = Jean-Pierre Luminet
  |author2-link=Jeffrey Weeks (mathematician)|first2=Jeff|last2= Weeks 
  |first3=Alain|last3= Riazuelo |first4=Roland|last4= Lehoucq 
  |first5=Jean-Phillipe |last5=Uzan
  | title = Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background
  | volume = 425
  | issue = 6958
  | pages = 593–595
  |journal = [[Nature (journal)|Nature]]
  | date = 2003-10-09
  | arxiv = astro-ph/0310253
  | issn = 
  | doi = 10.1038/nature01944
  | id = 
  | pmid = 14534579
  | bibcode=2003Natur.425..593L| s2cid = 4380713
 }}</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
  | last1 =Roukema
  | first1 =Boudewijn
  |first2=Zbigniew|last2= Buliński |first3=Agnieszka|last3= Szaniewska 
  |first4=Nicolas E.|last4= Gaudin
   | title =A test of the Poincare dodecahedral space topology hypothesis with the WMAP CMB data
  | journal = Astronomy and Astrophysics
  | volume =482
  | issue =3
  | pages =747–753
  | year = 2008
  | arxiv =0801.0006
  | doi =10.1051/0004-6361:20078777
  | id =
| bibcode=2008A&A...482..747L| s2cid =1616362
 }}</ref>
Data analysis from the [[Planck (spacecraft)|Planck spacecraft]] suggests that there is no observable non-trivial topology to the universe.<ref>Planck Collaboration, "[https://arxiv.org/abs/1502.01593 Planck 2015 results. XVIII. Background geometry & topology]", (2015) ArXiv 1502.01593</ref>