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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&nbsp;''n''&nbsp;≥&nbsp;1. That is, 
 
:''H''<sub>0</sub>(''X'','''Z''') = '''Z''' = ''H''<sub>''n''</sub>(''X'','''Z''')
 
and
 
:''H''<sub>''i''</sub>(''X'','''Z''') = {0} for all other ''i''.
 
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]]).
 
A '''rational homology sphere''' is defined similarly but using homology with rational coefficients.
 
==Poincaré homology sphere==
<!-- Henri Poincaré links here -->
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.
 
===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|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 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>.
 
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">[http://physicsworld.com/cws/article/news/18368 "Is the universe a dodecahedron?"], article at PhysicsWorld.</ref><ref name="Nat03">{{cite journal
  | last = Luminet
  | first = Jean-Pierre
  | authorlink = Jean-Pierre Luminet
  | coauthors = [[Jeffrey Weeks (mathematician)|Jeff Weeks]], Alain Riazuelo, Roland Lehoucq, Jean-Phillipe Uzan
  | title = Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background
  | journal = Nature
  | volume = 425
  | issue = 6958
  | pages = 593–595
  | publisher = [[Nature]]
  | location =
  | date = 2003-10-09
  | arxiv = astro-ph/0310253
  | issn =
  | doi = 10.1038/nature01944
  | id =
  | pmid = 14534579
  | 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
  | last =Roukema
  | first =Boudewijn
  | authorlink =
  | coauthors = Zbigniew Buliński, Agnieszka Szaniewska, Nicolas E. 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
  | publisher =
  | year = 2008
  | arxiv =0801.0006
  | doi =10.1051/0004-6361:20078777
  | id =
| bibcode=2008A&A...482..747L}}</ref>
However, there is no strong support for the correctness of the model, as yet.
 
==Constructions and examples==
 
*Surgery on a knot in the 3-sphere ''S''<sup>3</sup> with framing +1 or &minus; 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 &minus;1.
 
*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.
 
*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''<sub>1</sub>, ..., ''a''<sub>''r''</sub> are integers all at least 2 such that any two are coprime. Then the [[Seifert fiber space]]
 
:: <math>\{b, (o_1,0);(a_1,b_1),\dots,(a_r,b_r)\}\,</math>
 
: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
 
:: <math>b+b_1/a_1+\cdots+b_r/a_r=1/(a_1\cdots a_r).</math>
 
:(There is always a way to choose the ''b''&prime;s, and the homology sphere does not depend (up to isomorphism) on the choice of ''b''&prime;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''&prime;s are 2, 3, and 5 this gives the Poincaré sphere. If there are at least 3 ''a''&prime;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''')]].
 
==Invariants==
*The [[Rokhlin invariant]] is a '''Z'''/2'''Z''' valued invariant of homology 3-spheres.
*The [[Casson invariant]] is an integer valued invariant of homology 3-spheres, whose reduction mod 2 is the Rokhlin invariant.
 
==Applications==
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.)
 
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
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.
 
==References==
{{Reflist}}
 
==Selected reading==
*  Emmanuel Dror, ''Homology spheres'', Israel Journal of Mathematics 15 (1973), 115&ndash;129. {{MathSciNet|0328926}}
* 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.&nbsp;1–34.
* [[Robion Kirby]], Martin Scharlemann, ''Eight faces of the Poincaré homology 3-sphere''. Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), pp.&nbsp;113&ndash;146, [[Academic Press]], New York-London, 1979.
* [[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&ndash;72.  {{MathSciNet|0253347}}
* 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
 
==External links==
*[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]]
*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 ]
*[http://physicsworld.com/cws/article/print/23009 A cosmic hall of mirrors] - physicsworld (26 Sep 2005)
 
[[Category:Topological spaces]]
[[Category:Homology theory]]
[[Category:3-manifolds]]
[[Category:Spheres]]

Revision as of 03:19, 19 November 2013

In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n ≥ 1. That is,

H0(X,Z) = Z = Hn(X,Z)

and

Hi(X,Z) = {0} for all other i.

Therefore X is a connected space, with one non-zero higher Betti number: bn. 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 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.

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 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 S3/Ĩ where Ĩ is the binary icosahedral group, the perfect double cover of I embedded in S3.

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.[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 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 5-sphere around 0 with this complex surface) 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 a1, ..., ar are integers all at least 2 such that any two are coprime. Then the Seifert fiber space
{b,(o1,0);(a1,b1),,(ar,br)}
over the sphere with exceptional fibers of degrees a1, ..., ar is a homology sphere, where the b's are chosen so that
b+b1/a1++br/ar=1/(a1ar).
(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).

Invariants

  • The Rokhlin invariant is a Z/2Z valued invariant of homology 3-spheres.
  • The Casson invariant is an integer valued invariant of homology 3-spheres, whose reduction mod 2 is the Rokhlin invariant.

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. Template:As of 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.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Selected reading

External links

  1. "Is the universe a dodecahedron?", article at PhysicsWorld.
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    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  3. One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang