1168:
showed 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 is not triangulable.
396:). 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
996:
170:
839:
563:. 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. Data analysis from the
1222:; Riazuelo, Alain; Lehoucq, Roland; Uzan, Jean-Phillipe (2003-10-09). "Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background".
342:
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.
583:
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.
232:
1087:
450:
1285:
Roukema, Boudewijn; Buliński, Zbigniew; Szaniewska, Agnieszka; Gaudin, Nicolas E. (2008). "A test of the
Poincare dodecahedral space topology hypothesis with the WMAP CMB data".
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479:
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85:
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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
1152:
Galewski and Stern showed that all compact topological manifolds (without boundary) of dimension at least 5 are homeomorphic to simplicial complexes
876:
93:
322:
The
Poincaré homology sphere (also known as Poincaré dodecahedral space) is a particular example of a homology sphere, first constructed by
544:
722:
1595:
1183:
658:
657:, and every homology 3-sphere can be written as a connected sum of prime homology 3-spheres in an essentially unique way. (See
354:. Each face of the dodecahedron is identified with its opposite face, using the minimal clockwise twist to line up the faces.
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1584:, Encyclopaedia of Mathematical Sciences, vol 140. Low-Dimensional Topology, I. Springer-Verlag, Berlin, 2002.
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1645:
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1024:′s, not 2, 3, 5, then this is an acyclic homology 3-sphere with infinite fundamental group that has a
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is at most 2 this is just the usual 3-sphere; otherwise they are distinct non-trivial homology spheres. If the
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is an integer valued invariant of homology 3-spheres, whose reduction mod 2 is the
Rokhlin invariant.
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610:= 0 (in other words, the intersection of a small 3-sphere around 0 with this complex surface) is a
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A 16-Vertex
Triangulation of the Poincaré Homology 3-Sphere and Non-PL Spheres with Few Vertices
1517:. Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), pp. 113–146,
358:
each pair of opposite faces together using this identification yields a closed 3-manifold. (See
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that is a topological manifold but not a PL manifold. (It is not a PL manifold because the
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1353:"Pin(2)-equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture"
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Geometric
Topology (Proceedings Georgia Topology Conference, Athens Georgia, 1977)
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1020:′s are 2, 3, and 5 this gives the PoincarĂ© sphere. If there are at least 3
598:
are pairwise relatively prime positive integers then the link of the singularity
1472:(1980). "Classification of simplicial triangulations of topological manifolds".
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401:
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to the 3-sphere. In this case, the
Poincaré homology sphere is isomorphic to
1398:(1979). "A universal 5-manifold with respect to simplicial triangulations".
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1109:
is a homology 3-sphere not homeomorphic to the standard 3-sphere, then the
567:
suggests that there is no observable non-trivial topology to the universe.
539:
In 2003, lack of structure on the largest scales (above 60 degrees) in the
527:. The Poincaré homology sphere results from +1 surgery on the right-handed
1238:
331:
43:
1255:
991:{\displaystyle b+b_{1}/a_{1}+\cdots +b_{r}/a_{r}=1/(a_{1}\cdots a_{r}).}
165:{\displaystyle H_{0}(X,\mathbb {Z} )=H_{n}(X,\mathbb {Z} )=\mathbb {Z} }
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Alternatively, the
Poincaré homology sphere can be constructed as the
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58:
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1352:
834:{\displaystyle \{b,(o_{1},0);(a_{1},b_{1}),\dots ,(a_{r},b_{r})\}\,}
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are integers all at least 2 such that any two are coprime. Then the
314:
is defined similarly but using homology with rational coefficients.
1338:
27:
Topological manifold whose homology coincides with that of a sphere
1369:
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373:
362:
for a similar construction, using more "twist", that results in a
1339:
Planck 2015 results. XVIII. Background geometry & topology
1164:ÎŁ#ÎŁ of ÎŁ with itself bounds a smooth acyclic 4-manifold.
630:). It is homeomorphic to the standard 3-sphere if one of
1530:"Smooth homology spheres and their fundamental groups"
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1141:. In other words, this gives an example of a finite
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with framing +1 or −1 gives a homology sphere.
400:
of SO(3) which can be realized as the group of unit
847:over the sphere with exceptional fibers of degrees
1129:is homeomorphic to the standard 5-sphere, but its
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350:A simple construction of this space begins with a
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1535:Transactions of the American Mathematical Society
330:, it is the only homology 3-sphere (besides the
1626:The best picture of Poincare's homology sphere
1515:Eight faces of the Poincaré homology 3-sphere
8:
1357:Journal of the American Mathematical Society
827:
726:
642:is 1, and Σ(2, 3, 5) is the Poincaré sphere.
221:
215:
1429:Dror, Emmanuel (1973). "Homology spheres".
227:{\displaystyle H_{i}(X,\mathbb {Z} )=\{0\}}
1082:{\displaystyle \mathbb {Z} /2\mathbb {Z} }
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1089:-valued invariant of homology 3-spheres.
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1149:of a point is not always a 4-sphere.)
614:that is a homology 3-sphere, called a
445:{\displaystyle S^{3}/{\widetilde {I}}}
1004:(There is always a way to choose the
547:spacecraft led to the suggestion, by
7:
1156:there is a homology 3 sphere ÎŁ with
388:and dodecahedron, isomorphic to the
703:{\displaystyle a_{1},\ldots ,a_{r}}
1133:(induced by some triangulation of
1028:modeled on the universal cover of
576:Surgery on a knot in the 3-sphere
25:
1549:10.1090/S0002-9947-1969-0253347-3
1203:"Is the universe a dodecahedron?"
1117:is an example of a 4-dimensional
1582:Invariants of Homology 3-Spheres
1184:Moore space (algebraic topology)
863:is a homology sphere, where the
659:Prime decomposition (3-manifold)
543:as observed for one year by the
474:{\displaystyle {\widetilde {I}}}
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1:
1432:Israel Journal of Mathematics
1351:Manolescu, Ciprian (2016).
1125:. The double suspension of
541:cosmic microwave background
250:, with one non-zero higher
18:Poincaré dodecahedral space
1672:
1341:", (2015) ArXiv 1502.01593
1317:10.1051/0004-6361:20078777
1287:Astronomy and Astrophysics
1205:, article at PhysicsWorld.
571:Constructions and examples
287:. It does not follow that
555:and colleagues, that the
1521:, New York-London, 1979.
483:binary icosahedral group
340:binary icosahedral group
318:Poincaré homology sphere
312:rational homology sphere
1337:Planck Collaboration, "
1309:2008A&A...482..747L
1179:Eilenberg–MacLane space
523:Another approach is by
280:{\displaystyle b_{n}=1}
80:{\displaystyle n\geq 1}
1513:, Martin Scharlemann,
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380:(i.e., the rotational
334:itself) with a finite
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1475:Annals of Mathematics
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557:shape of the universe
553:Observatoire de Paris
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513:{\displaystyle S^{3}}
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364:hyperbolic 3-manifold
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1406:. pp. 345–350.
1216:Luminet, Jean-Pierre
1123:topological manifold
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867:s are chosen so that
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328:spherical 3-manifold
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1402:. New York-London:
1256:10.1038/nature01944
1248:2003Natur.425..593L
712:Seifert fiber space
549:Jean-Pierre Luminet
360:Seifert–Weber space
61:, for some integer
1641:Topological spaces
1580:Nikolai Saveliev,
1445:10.1007/BF02764597
1143:simplicial complex
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988:
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612:Brieskorn manifold
510:
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376:/I where I is the
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32:algebraic topology
1468:Galewski, David;
1394:Galewski, David;
1232:(6958): 593–595.
1166:Ciprian Manolescu
1158:Rokhlin invariant
1119:homology manifold
1052:Rokhlin invariant
1026:Thurston geometry
565:Planck spacecraft
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439:
390:alternating group
378:icosahedral group
336:fundamental group
297:fundamental group
16:(Redirected from
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1526:Kervaire, Michel
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1423:Selected reading
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1602:External links
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1519:Academic Press
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1439:(2): 115–129.
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1651:3-manifolds
1620:Lecture by
1482:(1): 1–34.
1363:: 147–176.
1220:Weeks, Jeff
1139:PL manifold
1137:) is not a
651:irreducible
618:3-sphere ÎŁ(
402:quaternions
386:icosahedron
326:. Being a
254:, namely,
87:. That is,
49:having the
1635:Categories
1190:References
1111:suspension
1045:Invariants
242:Therefore
1542:: 67–72.
1461:189796498
1370:1303.2354
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616:Brieskorn
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437:~
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332:3-sphere
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