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,
and
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.
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.
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 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.
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] Data analysis from the Planck spacecraft suggests that there is no observable non-trivial topology to the universe.[4]
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. Ciprian Manolescu showed[5] 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[6] is not triangulable.
American-Belgian semi-automatic pistol This article possibly contains original research. Please improve it by verifying the claims made and adding inline citations. Statements consisting only of original research should be removed. (October 2022) (Learn how and when to remove this template message) Browning Hi-Power Fabrique Nationale Browning Hi-PowerTypeSemi-automatic pistolPlace of origin Belgium United States Service historyIn service1935–present[1]Used bySee Us…
The World's Most Livable Cities adalah nama informal yang diberikan pada daftar kota dalam survei tahunan kondisi hidup. Dua yang terkenal[butuh rujukan] adalah Mercer Quality of Living Survey dan World's Most Livable Cities dari The Economist. Quality of Living Survey The Economist's World's Most Liveable Cities 2011 (Top 10)[1] Kota Negara Rating 1 Melbourne Australia 97.5 2 Vienna Austria 97.4 3 Vancouver Kanada 97.3 4 Toronto Kanada 97.2 5 Calgary …
SingrauliSuper Thermal Power StationCountryIndiaLocationShaktinagar, Uttar PradeshCoordinates24°06′19″N 82°42′23″E / 24.105168°N 82.706384°E / 24.105168; 82.706384StatusOperationalCommission date1986Owner(s)NTPCThermal power stationPrimary fuelCoalPower generation Units operationalStage - I: 5 × 200 MW Stage -II: 2 × 500 MW Solar PV: 15 MWNameplate capacity2,015 MWExternal linksWebsitewww.ntpc.co.in[edit on Wikidat…
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Vittorio MetzLahir(1904-07-18)18 Juli 1904Roma, ItaliaMeninggal11 Maret 1984(1984-03-11) (umur 79)Roma, ItaliaPekerjaanPenulis naskah, sutradaraTahun aktif1914-1965 Vittorio Metz (18 Juli 1904 – 1 Maret 1984) adalah seorang penulis naskah dan sutradara asal Italia. Ia menulis untuk 113 film antara 1939 dan 1977.[1] Filmografi pilihan Defendant, Stand Up! (1939) Lo vedi come sei... lo vedi come sei? (1939) The Pirate's Dream (1940) Non me lo dire! (1940) Annabell…
American explorer, writer and photographer Harriet Chalmers AdamsBorn(1875-10-22)October 22, 1875Stockton, California, USDiedJuly 17, 1937(1937-07-17) (aged 61)Nice, FranceResting placeChapel of the ChimesOccupation(s)Explorer, writer, photographerSpouseFranklin Pierce Adams Harriet Chalmers Adams (October 22, 1875 – July 17, 1937[1]) was an American explorer, writer and photographer. She traveled extensively in South America, Asia and the South Pacific in the early 20th century, …
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You can help expand this article with text translated from the corresponding article in French. (November 2012) Click [show] for important translation instructions. View a machine-translated version of the French article. Machine translation, like DeepL or Google Translate, is a useful starting point for translations, but translators must revise errors as necessary and confirm that the translation is accurate, rather than simply copy-pasting machine-translated text into the English Wikipedi…
Shopping mall in Ohio, United StatesDayton MallLocationMiami Township, Montgomery County, Ohio, United StatesCoordinates39°38′02″N 84°13′14″W / 39.63395°N 84.22060°W / 39.63395; -84.22060Opening dateFebruary 12, 1970; 53 years ago (1970-02-12)DeveloperEdward J. DeBartolo CorporationManagementSpinoso Real Estate GroupNo. of stores and services158No. of anchor tenants13 (11 open, 2 vacant)Total retail floor area1,416,846 square feet (131,629…
Este artigo não cita fontes confiáveis. Ajude a inserir referências. Conteúdo não verificável pode ser removido.—Encontre fontes: ABW • CAPES • Google (N • L • A) (Janeiro de 2013) Manuel Álvarez Zamora Nascimento 1800Villa de Álvarez Morte 26 de agosto de 1857 Cidadania México Ocupação político, militar [edite no Wikidata] Estátua do General Manuel Álvarez no México José Manuel Dolores Álvarez Zamora (Colima,…
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Herrsching a.AmmerseeLittle castle Lambang kebesaranLetak Herrsching a.Ammersee di Starnberg NegaraJermanNegara bagianBayernWilayahOberbayernKreisStarnbergSubdivisions3 OrtsteilePemerintahan • MayorChristian Schiller (Ein Bürgermeister für Alle)Luas • Total20,88 km2 (806 sq mi)Ketinggian568 m (1,864 ft)Populasi (2013-12-31)[1] • Total10.213 • Kepadatan4,9/km2 (13/sq mi)Zona waktuWET/WMPET (UTC+1/+2)Kode …
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