2360:
31:
2743:
1536:
625:
2636:, then gluing the faces together in pairs using affine-linear maps. Label the vertices 0, 1, 2, 3. Glue the face with vertices 0,1,2 to the face with vertices 3,1,0 in that order. Glue the face 0,2,3 to the face 3,2,1 in that order. In the hyperbolic structure of the Gieseking manifold, this ideal tetrahedron is the canonical polyhedral decomposition of
4508:). In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by William
4300:
should be irreducible and atoroidal are necessary, as hyperbolic manifolds have these properties. However the condition that the manifold be Haken is unnecessarily strong. Thurston's hyperbolization conjecture states that a closed irreducible atoroidal 3-manifold with infinite fundamental group is
168:
Phenomena in three dimensions can be strikingly different from phenomena in other dimensions, and so there is a prevalence of very specialized techniques that do not generalize to dimensions greater than three. This special role has led to the discovery of close connections to a diversity of other
2554:
to its opposite in a way that produces a closed 3-manifold. There are three ways to do this gluing consistently. Opposite faces are misaligned by 1/10 of a turn, so to match them they must be rotated by 1/10, 3/10 or 5/10 turn; a rotation of 3/10 gives the
Seifert–Weber space. Rotation of 1/10
797:
4647:
implicitly implied (by referring to a then unpublished longer manuscript) that he had proven the
Virtually fibered conjecture for the case where the 3-manifold is closed, hyperbolic, and Haken. This was followed by a survey article in Electronic Research Announcements in Mathematical Sciences.
1247:
338:
238:
contributions to the theory allow one to also consider, in many cases, the additional structure given by a particular
Thurston model geometry (of which there are eight). The most prevalent geometry is hyperbolic geometry. Using a geometry in addition to special surfaces is often fruitful.
4199:
Another important result by
Thurston is that volume decreases under hyperbolic Dehn filling. In fact, the theorem states that volume decreases under topological Dehn filling, assuming of course that the Dehn-filled manifold is hyperbolic. The proof relies on basic properties of the
2505:
is a
Poincaré sphere. 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. However, there is no strong support for the correctness of the model, as yet.
1912:
2932:, where they can be split up into 3-balls along incompressible surfaces. Haken also showed that there was a finite procedure to find an incompressible surface if the 3-manifold had one. Jaco and Oertel gave an algorithm to determine if a 3-manifold was Haken.
4664:
and results of Wise in proving the
Malnormal Special Quotient Theorem and results of Bergeron and Wise for the cubulation of groups. Taken together with Wise's results, this implies the virtually fibered conjecture for all closed hyperbolic 3-manifolds.
3014:
of a 3-manifold with the property that there is a single transverse circle intersecting every leaf. By transverse circle, is meant a closed loop that is always transverse to the tangent field of the foliation. Equivalently, by a result of
3500:
states that if a pair of disjoint simple closed curves on the boundary of a three manifold are freely homotopic then they cobound a properly embedded annulus. This should not be confused with the high dimensional theorem of the same name.
2575:
there exist regular dodecahedra with any dihedral angle between 60° and 117°, and the hyperbolic dodecahedron with dihedral angle 72° may be used to give the
Seifert–Weber space a geometric structure as a hyperbolic manifold. It is a
1531:{\displaystyle {\begin{aligned}H_{1}(M)&=H_{1}(M_{1})\oplus \cdots \oplus H_{1}(M_{n})\\H_{2}(M)&=H_{2}(M_{1})\oplus \cdots \oplus H_{2}(M_{n})\\\pi _{1}(M)&=\pi _{1}(M_{1})*\cdots *\pi _{1}(M_{n})\end{aligned}}}
660:
620:{\displaystyle {\begin{aligned}H_{0}(M)&=H^{3}(M)=&\mathbb {Z} \\H_{1}(M)&=H^{2}(M)=&\pi /\\H_{2}(M)&=H^{1}(M)=&{\text{Hom}}(\pi ,\mathbb {Z} )\\H_{3}(M)&=H^{0}(M)=&\mathbb {Z} \end{aligned}}}
4774:
6014:
3269:
2984:. This contrasts strongly with higher-dimensional manifolds which need not admit smooth or piecewise linear structures. Assuming smoothness the existence of a Heegaard splitting also follows from the work of
2566:
With the 3/10-turn gluing pattern, the edges of the original dodecahedron are glued to each other in groups of five. Thus, in the
Seifert–Weber space, each edge is surrounded by five pentagonal faces, and the
1775:
38:. All of the cubes in the image are the same cube, since light in the manifold wraps around into closed loops, the effect is that the cube is tiling all of space. This space has finite volume and no boundary.
3718:'s theorems on topological rigidity say that certain 3-manifolds (such as those with an incompressible surface) are homeomorphic if there is an isomorphism of fundamental groups which respects the boundary.
2288:
4887:
and outlined in a talk August 4, 2009 at the FRG (Focused
Research Group) Conference hosted by the University of Utah. A preprint appeared on the arxiv in October 2009. Their paper was published in the
5329:
Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, pp. 71–84, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978.
5131:; 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".
4114:
1064:
1252:
665:
343:
3508:
is as follows: Let M be a compact, irreducible 3-manifold with nonempty boundary. If M admits an essential map of a torus, then M admits an essential embedding of either a torus or an annulus
2914:. Sometimes one considers only orientable Haken manifolds, in which case a Haken manifold is a compact, orientable, irreducible 3-manifold that contains an orientable, incompressible surface.
1786:
1969:, the curves intersect each other orthogonally (in the yellow points) as in 4D. All curves are circles: the curves that intersect <0,0,0,1> have infinite radius (= straight line).
4050:
3944:
1955:
1192:
4703:
2075:
1694:
258:
3-manifolds are an interesting special case of low-dimensional topology because their topological invariants give a lot of information about their structure in general. If we let
165:, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.
987:
4247:
5194:
Roukema, Boudewijn; Zbigniew Buliński; Agnieszka
Szaniewska; Nicolas E. Gaudin (2008). "A test of the Poincare dodecahedral space topology hypothesis with the WMAP CMB data".
4800:
4219:
are taken to nontrivial limits in the set of volumes. In fact, one can further conclude, as did Thurston, that the set of volumes of finite volume hyperbolic 3-manifolds has
3310:
318:
1645:
2116:
1602:
1140:
932:
893:
842:
4156:
3450:
3414:
3486:
3333:
4832:
2666:
4947:
4183:
3974:
3705:
2038:
1239:
2011:
1007:
652:
78:) to a small and close enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.
4970:
2948:
where every leaf is incompressible and end incompressible, if the complementary regions of the lamination are irreducible, and if there are no spherical leaves.
4460:
to systematically excise singular regions as they develop, in a controlled way. Several teams of mathematicians have verified that Perelman's proof is correct.
3374:
1559:
1212:
1088:
276:
139:
112:
2874:
4864:
has a surface subgroup. By "surface subgroup" we mean the fundamental group of a closed surface not the 2-sphere. This problem is listed as Problem 3.75 in
1993:
that forms the boundary of a ball in four dimensions. Many examples of 3-manifolds can be constructed by taking quotients of the 3-sphere by a finite group
2625:
and has the smallest volume among non-compact hyperbolic manifolds, having volume approximately 1.01494161. It was discovered by Hugo Gieseking (
1573:
For the case of a 3-manifold given by a connected sum of prime 3-manifolds, it turns out there is a nice description of the second fundamental group as a
792:{\displaystyle {\begin{aligned}H_{1}(\pi ;\mathbb {Z} )&\cong \pi /\\H^{1}(\pi ;\mathbb {Z} )&\cong {\text{Hom}}(\pi ,\mathbb {Z} )\end{aligned}}}
1936:
Euclidean 3-space is the most important example of a 3-manifold, as all others are defined in relation to it. This is just the standard 3-dimensional
4549:
are corollaries of the geometrization conjecture, although there are shorter proofs of the former that do not lead to the geometrization conjecture.
3579:
4425:, the theorem concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary (a
3726:
Waldhausen conjectured that every closed orientable 3-manifold has only finitely many Heegaard splittings (up to homeomorphism) of any given genus.
6115:
4531:
satisfy the geometrization conjecture. Thurston announced a proof in the 1980s and since then several complete proofs have appeared in print.
2668:. The triangulation has one tetrahedron, two faces, one edge and no vertices, so all the edges of the original tetrahedron are glued together.
2428:) that have a constant positive curvature. When embedded to a Euclidean space (of a higher dimension), every point of a hyperbolic space is a
5586:
246:
of 3-manifolds strongly reflect the geometric and topological information belonging to a 3-manifold. Thus, there is an interplay between
5403:
4708:
4196:. Troels Jorgensen's study of the geometric topology further shows that all nontrivial limits arise by Dehn filling as in the theorem.
3197:
1699:
5364:
2851:
2222:
4648:
Several more preprints have followed, including the aforementioned longer manuscript by Wise. In March 2012, during a conference at
2865:, which belongs to the even-dimensional world. Both contact and symplectic geometry are motivated by the mathematical formalism of
4469:
2571:
between these pentagons is 72°. This does not match the 117° dihedral angle of a regular dodecahedron in Euclidean space, but in
2364:
4917:
The cabling conjecture states that if Dehn surgery on a knot in the 3-sphere yields a reducible 3-manifold, then that knot is a
4293:
implies that if a manifold of dimension at least 3 has a hyperbolic structure of finite volume, then it is essentially unique.
6088:
4063:
1012:
5756:
3138:
if it cannot be presented as a connected sum of more than one manifold, none of which is the sphere of the same dimension.
2839:
2490:
4351:
2854:, one recognizes the condition as the opposite of the condition that the distribution be determined by a codimension one
1907:{\displaystyle \pi _{2}(M)={\frac {\mathbb {Z} \{\sigma _{1},\ldots ,\sigma _{n}\}}{(\sigma _{1}+\cdots +\sigma _{n})}}}
1962:
4434:
6130:
5953:
5924:
5895:
5862:
5836:
5807:
5678:
Kahn, Jeremy; Markovic, Vladimir (2009). "Immersing almost geodesic surfaces in a closed hyperbolic three manifold".
5619:
Kahn, Jeremy; Markovic, Vladimir (2009). "Immersing almost geodesic surfaces in a closed hyperbolic three manifold".
5347:
4545:. There are now several different manuscripts (see below) with details of the proof. The Poincaré conjecture and the
4282:
is a compact irreducible atoroidal Haken manifold whose boundary has zero Euler characteristic, then the interior of
2401:
55:
5917:
The Knot Book. An elementary introduction to the mathematical theory of knots. Revised reprint of the 1994 original.
3350:) gives conditions for elements of the second homotopy group of a 3-manifold to be represented by embedded spheres.
5256:
3099:
2980:
three-manifold may be so obtained; this follows from deep results on the triangulability of three-manifolds due to
2462:
homology sphere (also known as Poincaré dodecahedral space) is a particular example of a homology sphere. Being a
190:
4546:
3987:
3881:
2796:
2726:
220:
embedded in it. One can choose the surface to be nicely placed in the 3-manifold, which leads to the idea of an
5447:
4602:
4254:
2581:
2316:(with the action being taken as vector addition). Equivalently, the 3-torus is obtained from the 3-dimensional
202:
4433:
in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere. An
4350:. It is one of the fundamental properties of geometrically infinite hyperbolic 3-manifolds, together with the
1142:. In fact, from general theorems in topology, we find for a three manifold with a connected sum decomposition
5791:
5128:
5002:
3655:
1145:
4676:
2043:
1653:
4558:
4371:
3803:
3185:
3167:
2850:, both of which satisfy a 'maximum non-degeneracy' condition called 'complete non-integrability'. From the
2345:
with multiplication). Group multiplication on the torus is then defined by coordinate-wise multiplication.
802:
From this information a basic homotopy theoretic classification of 3-manifolds can be found. Note from the
5640:
Kahn, Jeremy; Markovic, Vladimir (2010). "Counting Essential Surfaces in a Closed Hyperbolic 3-Manifold".
2453:
6135:
5912:
5507:
4847:
4661:
4660:
for closed hyperbolic 3-manifolds. The proof built on results of Kahn and Markovic in their proof of the
4644:
2486:
142:
17:
5948:, Colloquium Publications, vol. 40, Providence, RI: American Mathematical Society, pp. x+238,
4429:
3-manifold). The Poincaré conjecture claims that if such a space has the additional property that each
2556:
937:
6005:
5887:
5587:
https://docs.google.com/file/d/0B45cNx80t5-2NTU0ZTdhMmItZTIxOS00ZGUyLWE0YzItNTEyYWFiMjczZmIz/edit?pli=1
5511:
4897:
4657:
4562:
4359:
4225:
3343:
3181:
3076:
2925:
asserts that every compact, irreducible 3-manifold with infinite fundamental group is virtually Haken.
2922:
2306:
4779:
4649:
3277:
281:
228:, or one can choose the complementary pieces to be as nice as possible, leading to structures such as
4872:
4355:
3538:
2702:
2577:
1607:
162:
5097:
4517:
4393:
4192:
and fundamental to the theory of hyperbolic 3-manifolds. It shows that nontrivial limits exist in
3616:
2475:
1958:
206:
4481:
each have a unique geometric structure that can be associated with them. It is an analogue of the
3150:
such that any collection of disjoint incompressible embedded surfaces of cardinality greater than
2080:
1576:
1105:
898:
850:
812:
4524:
4304:
4290:
4273:
4119:
3870:
3640:
3419:
3383:
2515:
320:
be its fundamental group, then a lot of information can be derived from them. For example, using
3864:
3458:
3315:
5092:
4482:
3781:
3707:
surgery coefficients. Furthermore, each component of the link can be assumed to be unknotted.
3552:
2911:
2433:
2359:
221:
174:
5442:
4513:
4418:
4412:
4301:
hyperbolic, and this follows from Perelman's proof of the Thurston geometrization conjecture.
2928:
Haken manifolds were introduced by Wolfgang Haken. Haken proved that Haken manifolds have a
2479:
5712:
5193:
4889:
4876:
4456:
to attack the problem. Perelman introduced a modification of the standard Ricci flow, called
4400:
4389:
4342:
The tameness theorem was conjectured by Marden. It was proved by Agol and, independently, by
4320:
3822:
2945:
2692:
2618:
2560:
2536:
2502:
2498:
2369:
2127:
63:
5217:
4805:
4399:
are determined by their topology together with certain "end invariants", which are geodesic
2643:
6140:
6105:
6023:
5963:
5934:
5905:
5872:
5846:
5817:
5567:
5537:
5470:
5424:
5385:
5279:
5213:
5152:
4920:
4857:
4621:
4586:
4161:
3952:
3757:
3715:
3687:
3639:
on fundamental groups. In particular, this means a finitely generated 3-manifold group is
2969:
is a decomposition of a compact oriented 3-manifold that results from dividing it into two
2721:
2712:
2463:
2016:
1977:. It consists of the set of points equidistant from a fixed central point in 4-dimensional
1217:
1996:
992:
637:
186:
8:
5247:
5124:
4893:
4617:
4505:
4449:
4310:
4208:
2866:
2862:
2778:
2637:
2494:
2405:
178:
6027:
5156:
4952:
4860:
states that the fundamental group of every closed, irreducible 3-manifold with infinite
3558:
such that each component of the 3-manifold obtained by cutting along the tori is either
321:
5739:
5721:
5679:
5641:
5620:
5599:
5487:
5398:
5229:
5203:
5176:
5142:
5069:
4501:
4497:
4486:
4444:
presented a proof of the conjecture in three papers made available in 2002 and 2003 on
4328:
3769:
3591:
3359:
3087:
3064:
3058:
2960:
2951:
Essential laminations generalize the incompressible surfaces found in Haken manifolds.
2758:
2604:
2592:
2441:
2417:
2354:
1986:
1982:
1544:
1197:
1073:
261:
229:
217:
210:
124:
97:
6054:
6009:
5311:
5294:
6084:
6059:
6041:
5993:
5949:
5920:
5891:
5858:
5832:
5803:
5343:
5168:
5073:
5061:
5022:
4884:
4861:
4625:
4594:
4478:
4396:
4324:
3857:
must intersect) is at most 1. Consequently, there are at most three Dehn fillings of
3601:
3517:
3020:
2847:
2770:
2471:
2421:
2397:
2393:
2137:
1931:
243:
150:
92:
51:
5988:
5971:
5760:
5743:
4949:-cable on some other knot, and the surgery must have been performed using the slope
4422:
2900:
2459:
6076:
6049:
6031:
5983:
5879:
5795:
5731:
5523:
5456:
5412:
5373:
5306:
5265:
5233:
5221:
5180:
5160:
5133:
5053:
5014:
4613:
4578:
4534:
4490:
4441:
4189:
3735:
3612:
3146:
Kneser-Haken finiteness says that for each compact 3-manifold, there is a constant
3080:
3031:
2826:
2585:
2572:
631:
325:
235:
115:
71:
5707:
5598:
Agol, Ian; Groves, Daniel; Manning, Jason (2012). "The virtual Haken conjecture".
5486:
Bergeron, Nicolas; Wise, Daniel T. (2009). "A boundary criterion for cubulation".
3646:
A simplified proof is given in, and a stronger uniqueness statement is proven in.
5959:
5930:
5901:
5868:
5842:
5813:
5664:
5533:
5466:
5420:
5381:
5275:
5225:
5111:
4582:
4493:
4430:
4426:
3795:
3761:
3665:
3563:
3177:
3024:
3016:
2835:
2806:
2786:
2766:
2746:
2687:
2533:
2529:
2409:
2385:
2380:
2310:
2211:
2199:
2149:
1978:
803:
118:
5735:
2547:. It is one of the first discovered examples of closed hyperbolic 3-manifolds.
5438:
5359:
4637:
4633:
4598:
4542:
4528:
4385:
4343:
4261:
4220:
3749:
3605:
3180:
and should more properly be called the "disk theorem". It was first proven by
3128:
3072:
2997:
2981:
2886:
2843:
2801:
2682:
2677:
2568:
2342:
329:
225:
198:
6080:
5377:
4552:
4305:
Tameness conjecture, also called the Marden conjecture or tame ends conjecture
3191:
A simple and useful version of the loop theorem states that if there is a map
6124:
6045:
5997:
5512:"Research announcement: The structure of groups with a quasiconvex hierarchy"
5461:
5270:
5251:
5065:
5026:
5018:
4575:
4572:
4417:
The 3-sphere is an especially important 3-manifold because of the now-proven
4381:
4336:
4332:
3865:
Thurston's hyperbolic Dehn surgery theorem and the Jørgensen–Thurston theorem
3799:
3791:
3632:
3544:
closed (i.e., compact and without boundary) 3-manifolds have a unique (up to
3541:
3124:
3117:
3113:
3109:
2896:
2716:
2622:
2331:
2324:
2175:
2145:
1966:
1099:
1067:
182:
75:
5416:
6063:
5824:
5660:
5528:
5327:
A new decomposition theorem for irreducible sufficiently-large 3-manifolds.
5172:
4865:
4776:
is not injective, then there exists a non-contractible simple closed curve
3834:
3673:
3575:
3571:
3163:
3046:
Some results are named as conjectures as a result of historical artifacts.
2731:
2588:
2551:
2485:
In 2003, lack of structure on the largest scales (above 60 degrees) in the
2429:
2376:
2165:
1937:
247:
194:
146:
6036:
5708:"Immersing almost geodesic surfaces in a closed hyperbolic three manifold"
2632:
The Gieseking manifold can be constructed by removing the vertices from a
1561:
which cannot be described as a connected sum of two 3-manifolds is called
6071:"Topologische Fragen der Differentialgeometrie 43. Gewebe und Gruppen ",
5699:
5147:
4880:
4835:
4347:
4267:
4201:
3677:
3636:
3620:
3531:
3035:
3007:
2870:
2633:
2610:
2521:
2338:
216:
A key idea in the theory is to study a 3-manifold by considering special
170:
43:
35:
5164:
4264:
has the smallest volume of any closed orientable hyperbolic 3-manifold.
5941:
5799:
5057:
4538:
4453:
4212:
3669:
3377:
2977:
2970:
2707:
2191:
5972:"Three dimensional manifolds, Kleinian groups and hyperbolic geometry"
3582:. The first two worked together, and the third worked independently.
2379:
meet at each edge, and eight meet at each vertex, like the cubes of a
5041:
4590:
4537:
sketched a proof of the full geometrization conjecture in 2003 using
3559:
3549:
3453:
3075:
in, states that any topological 3-manifold has an essentially unique
3011:
2855:
2774:
2334:
2195:
1990:
1647:
is infinite but not cyclic, if we take based embeddings of a 2-sphere
30:
1954:
5919:, Providence, RI: American Mathematical Society, pp. xiv+307,
5703:
5089:
On the Classification of Finite Groups Acting on Homology 3-Spheres
4769:{\displaystyle f_{\star }\colon \pi _{1}(S)\rightarrow \pi _{1}(T)}
4653:
3753:
3681:
3545:
3416:
is not the trivial group. Then there exists a non-zero element of
2861:
Contact geometry is in many ways an odd-dimensional counterpart of
2762:
2697:
2467:
2425:
2327:
1949:
59:
5726:
5684:
5646:
5625:
5604:
5492:
5208:
4207:
Jørgensen also showed that the volume function on this space is a
3264:{\displaystyle f\colon (D^{2},\partial D^{2})\to (M,\partial M)\,}
2444:
with respect to the radius of the ball, rather than polynomially.
1770:{\displaystyle \sigma _{i}(S^{2})\subset M_{i}-\{B^{3}\}\subset M}
4380:, originally conjectured by William Thurston and later proven by
2908:
2742:
2301:
under integral shifts in any coordinate. That is, the 3-torus is
3710:
2917:
A 3-manifold finitely covered by a Haken manifold is said to be
2792:
The following examples are particularly well-known and studied.
2640:
and Robert C. Penner. Moreover, the angle made by the faces is
1981:. Just as an ordinary sphere (or 2-sphere) is a two-dimensional
4901:
4879:. A proof of this case was announced in the Summer of 2009 by
3721:
2437:
2283:{\displaystyle \mathbf {T} ^{3}=S^{1}\times S^{1}\times S^{1}.}
1974:
1965:(blue) and hypermeridians (green). Because this projection is
67:
4215:
function. Thus by the previous results, nontrivial limits in
3845:(the minimal number of times that two simple closed curves in
2341:
is a compact abelian Lie group (when identified with the unit
2216:
The 3-dimensional torus is the product of 3 circles. That is:
5252:"Euclidean decompositions of noncompact hyperbolic manifolds"
4445:
3811:
3555:
3121:
2985:
2179:
5342:
Lecture Notes in Mathematics, 761. Springer, Berlin, 1979.
5123:
4553:
Virtually fibered conjecture and Virtually Haken conjecture
2815:
2413:
2317:
5516:
Electronic Research Announcements in Mathematical Sciences
3030:
Taut foliations were brought to prominence by the work of
4892:
in 2012. In June 2012, Kahn and Markovic were given the
4278:
One form of Thurston's geometrization theorem states: If
3837:
have cyclic fundamental group, then the distance between
2323:
A 3-torus in this sense is an example of a 3-dimensional
5568:
http://comet.lehman.cuny.edu/behrstock/cbms/program.html
4834:
is homotopically trivial. This conjecture was proven by
3619:
fundamental group, there is a compact three-dimensional
2671:
1989:
in three dimensions, a 3-sphere is an object with three
4463:
5401:; Swarup, Gadde A. (1990), "On Scott's core theorem",
4980:
The fundamental group of any finite volume hyperbolic
4286:
has a complete hyperbolic structure of finite volume.
4268:
Thurston's hyperbolization theorem for Haken manifolds
4109:{\displaystyle p_{i}^{2}+q_{i}^{2}\rightarrow \infty }
3019:, a codimension 1 foliation is taut if there exists a
1780:
then the second fundamental group has the presentation
1059:{\displaystyle \zeta _{M}\in H_{3}(\pi ,\mathbb {Z} )}
5340:
Homotopy equivalences of 3-manifolds with boundaries.
4955:
4923:
4808:
4782:
4711:
4679:
4512:, and implies several other conjectures, such as the
4228:
4164:
4122:
4066:
3990:
3955:
3884:
3690:
3672:, connected 3-manifold may be obtained by performing
3461:
3422:
3386:
3362:
3335:, then there is an embedding with the same property.
3318:
3280:
3200:
2869:, where one can consider either the even-dimensional
2646:
2225:
2083:
2046:
2019:
1999:
1789:
1702:
1656:
1610:
1579:
1547:
1250:
1220:
1200:
1148:
1108:
1076:
1015:
995:
940:
901:
853:
815:
663:
640:
341:
284:
264:
156:
127:
100:
5583:
The structure of groups with a quasiconvex hierarchy
4440:
After nearly a century of effort by mathematicians,
4260:
Also, Gabai, Meyerhoff & Milley showed that the
4253:. Further work characterizing this set was done by
2812:
The classes are not necessarily mutually exclusive.
1920:
1917:
giving a straightforward computation of this group.
847:
If we take the pushforward of the fundamental class
253:
4437:has been known in higher dimensions for some time.
2182:, hence admits a group structure; the covering map
5577:
5575:
4964:
4941:
4826:
4794:
4768:
4697:
4403:on some surfaces in the boundary of the manifold.
4241:
4177:
4150:
4108:
4044:
3968:
3938:
3699:
3480:
3444:
3408:
3368:
3327:
3304:
3263:
2660:
2282:
2110:
2069:
2032:
2005:
1906:
1769:
1688:
1639:
1596:
1553:
1530:
1233:
1206:
1186:
1134:
1082:
1058:
1001:
981:
926:
887:
836:
791:
646:
619:
312:
270:
133:
106:
6004:
5857:, Providence, RI: American Mathematical Society,
5831:, Providence, RI: American Mathematical Society,
5597:
5443:"The uniqueness of compact cores for 3-manifolds"
5299:Transactions of the American Mathematical Society
3347:
2466:, it is the only homology 3-sphere (besides the
1973:A 3-sphere is a higher-dimensional analogue of a
6122:
5551:A combination theorem for special cube complexes
5397:
4705:is a map of closed connected surfaces such that
3600:is a theorem about the finite presentability of
630:where the last two groups are isomorphic to the
6015:Proceedings of the National Academy of Sciences
5572:
5362:(1973), "Compact submanifolds of 3-manifolds",
4365:
2988:about handle decompositions from Morse theory.
2907:, meaning that it contains a properly embedded
2454:Homology sphere § Poincaré homology sphere
232:, which are useful even in the non-Haken case.
6010:"On Dehn's Lemma and the Asphericity of Knots"
4841:
3093:
2873:of a mechanical system or the odd-dimensional
2737:
2447:
1066:gives a complete algebraic description of the
5976:Bulletin of the American Mathematical Society
5246:
4975:
4636:with a finite-to-one covering map) that is a
3711:Waldhausen's theorems on topological rigidity
3491:
3086:As corollary, every compact 3-manifold has a
1604:-module. For the special case of having each
205:. 3-manifold theory is considered a part of
5698:
5677:
5639:
5618:
5485:
5481:
5479:
4448:. The proof followed on from the program of
3722:Waldhausen conjecture on Heegaard splittings
3649:
2858:on the manifold ('complete integrability').
1861:
1829:
1758:
1745:
5436:
5404:Bulletin of the London Mathematical Society
5295:"On the torus theorem and its applications"
5105:
4045:{\displaystyle M(u_{1},u_{2},\dots ,u_{n})}
3939:{\displaystyle M(u_{1},u_{2},\dots ,u_{n})}
3157:
3106:prime decomposition theorem for 3-manifolds
2559:, and rotation by 5/10 gives 3-dimensional
2550:It is constructed by gluing each face of a
1098:One important topological operation is the
6073:Gesammelte Abhandlungen / Collected Papers
5365:Journal of the London Mathematical Society
5007:Journal of the London Mathematical Society
4643:In a posting on the ArXiv on 25 Aug 2009,
3876:Thurston's hyperbolic Dehn surgery theorem
3534:construct given by the following theorem:
3141:
2482:cannot be stated in homology terms alone.
2190:is a map of groups Spin(3) → SO(3), where
2121:
6053:
6035:
5987:
5725:
5683:
5645:
5624:
5603:
5557:Finiteness properties of cubulated groups
5527:
5491:
5476:
5460:
5310:
5269:
5207:
5146:
5096:
4668:
4624:three-dimensional manifold with infinite
4158:corresponding to non-empty Dehn fillings
3946:is hyperbolic as long as a finite set of
3775:
3260:
3184:in 1956, along with Dehn's lemma and the
2834:is the study of a geometric structure on
2626:
2595:by dodecahedra with this dihedral angle.
2474:. Its fundamental group is known as the
2140:of lines passing through the origin 0 in
1816:
1581:
1568:
1049:
778:
749:
688:
609:
547:
397:
5969:
5878:
4907:
4875:, the only open case was that of closed
4509:
3608:. The precise statement is as follows:
2816:Some important structures on 3-manifolds
2741:
2358:
1953:
29:
5884:Three-dimensional geometry and topology
5852:
5292:
3041:
2935:
2320:by gluing the opposite faces together.
1187:{\displaystyle M=M_{1}\#\cdots \#M_{n}}
1009:together with the group homology class
14:
6123:
5785:
5086:
5039:
5000:
4698:{\displaystyle f\colon S\rightarrow T}
4496:can be given one of three geometries (
4477:states that certain three-dimensional
4406:
3049:We begin with the purely topological:
2509:
2432:. Another distinctive property is the
2337:. This follows from the fact that the
2070:{\displaystyle \pi \to {\text{SO}}(4)}
1961:of the hypersphere's parallels (red),
1689:{\displaystyle \sigma _{i}:S^{2}\to M}
5946:The Geometric Topology of 3-Manifolds
5911:
5358:
5187:
4912:
3585:
2954:
2672:Some important classes of 3-manifolds
2598:
2493:spacecraft led to the suggestion, by
2348:
2330:. It is also an example of a compact
5940:
5823:
5666:Problems in low-dimensional topology
5506:
5117:
5042:"On embedded spheres in 3-manifolds"
4984:-manifold does not have Property Ď„.
4632:. That is, it has a finite cover (a
4475:Thurston's geometrization conjecture
4470:Thurston's geometrization conjecture
4464:Thurston's geometrization conjecture
3611:Given a 3-manifold (not necessarily
3511:
2621:3-manifold of finite volume. It is
2440:in hyperbolic 3-space: it increases
1925:
6103:
5829:Lectures on three-manifold topology
4352:density theorem for Kleinian groups
3729:
3548:) minimal collection of disjointly
3452:having a representative that is an
2820:
2478:and has order 120. This shows the
24:
6107:Notes on basic 3-manifold topology
5779:
4103:
3319:
3289:
3251:
3223:
3052:
2297:can be described as a quotient of
1171:
1165:
1119:
982:{\displaystyle \zeta _{M}=q_{*}()}
157:Mathematical theory of 3-manifolds
25:
6152:
6116:A Bestiary of Topological Objects
6097:
5312:10.1090/s0002-9947-1976-0394666-3
5112:"Is the universe a dodecahedron?"
4296:The conditions that the manifold
4242:{\displaystyle \omega ^{\omega }}
2991:
2880:
2877:that includes the time variable.
1921:Important examples of 3-manifolds
1093:
254:Invariants describing 3-manifolds
56:three-dimensional Euclidean space
5325:Jaco, William; Shalen, Peter B.
5257:Journal of Differential Geometry
5040:Swarup, G. Ananda (1973-06-01).
4795:{\displaystyle \alpha \subset S}
3305:{\displaystyle f|\partial D^{2}}
3154:must contain parallel elements.
3100:prime decomposition (3-manifold)
2541:Seifert–Weber dodecahedral space
2489:as observed for one year by the
2228:
2212:Torus § n-dimensional torus
313:{\displaystyle \pi =\pi _{1}(M)}
191:topological quantum field theory
62:can be thought of as a possible
5989:10.1090/s0273-0979-1982-15003-0
5757:"2012 Clay Research Conference"
5749:
5692:
5671:
5654:
5633:
5612:
5591:
5543:
5500:
5430:
5391:
5352:
5332:
4547:spherical space form conjecture
4388:, and Yair Minsky, states that
4249:. This result is known as the
3861:with cyclic fundamental group.
2727:Surface bundles over the circle
2396:that can be characterized by a
1640:{\displaystyle \pi _{1}(M_{i})}
6006:Papakyriakopoulos, Christos D.
5448:Pacific Journal of Mathematics
5319:
5286:
5240:
5080:
5033:
5003:"On a Theorem of C. B. Thomas"
4994:
4936:
4924:
4814:
4763:
4757:
4744:
4741:
4735:
4689:
4603:surface bundle over the circle
4100:
4039:
3994:
3933:
3888:
3472:
3439:
3433:
3403:
3397:
3353:One example is the following:
3285:
3257:
3242:
3239:
3236:
3207:
2785:is a hyperbolic link with one
2582:order-5 dodecahedral honeycomb
2363:A perspective projection of a
2064:
2058:
2050:
1898:
1866:
1826:
1820:
1806:
1800:
1726:
1713:
1680:
1634:
1621:
1591:
1585:
1521:
1508:
1486:
1473:
1453:
1447:
1430:
1417:
1395:
1382:
1362:
1356:
1339:
1326:
1304:
1291:
1271:
1265:
1053:
1039:
976:
973:
967:
964:
921:
912:
882:
876:
860:
854:
825:
782:
768:
753:
739:
722:
710:
692:
678:
600:
594:
574:
568:
551:
537:
524:
518:
498:
492:
475:
463:
447:
441:
421:
415:
388:
382:
362:
356:
307:
301:
203:partial differential equations
13:
1:
5970:Thurston, William P. (1982).
5792:American Mathematical Society
5563:Cubulating malnormal amalgams
4987:
4656:announced he could prove the
2545:hyperbolic dodecahedral space
1985:that forms the boundary of a
86:
81:
5250:; Penner, Robert C. (1988).
4569:virtually fibered conjecture
4559:Virtually fibered conjecture
4421:. Originally conjectured by
4372:Ending lamination conjecture
4366:Ending lamination conjecture
3744:(now proven) states that if
3168:Sphere theorem (3-manifolds)
2132:Real projective 3-space, or
2111:{\displaystyle M=S^{3}/\pi }
1597:{\displaystyle \mathbb {Z} }
1135:{\displaystyle M_{1}\#M_{2}}
927:{\displaystyle H_{3}(B\pi )}
888:{\displaystyle \in H_{3}(M)}
837:{\displaystyle q:M\to B\pi }
7:
5736:10.4007/annals.2012.175.3.4
5293:Feustel, Charles D (1976).
4854:surface subgroup conjecture
4848:Surface subgroup conjecture
4842:Surface subgroup conjecture
4662:Surface subgroup conjecture
4319:states that every complete
4151:{\displaystyle p_{i}/q_{i}}
3445:{\displaystyle \pi _{2}(M)}
3409:{\displaystyle \pi _{2}(M)}
3094:Prime decomposition theorem
2738:Hyperbolic link complements
2532:and Constantin Weber) is a
2487:cosmic microwave background
2448:Poincaré dodecahedral space
2408:. It is distinguished from
1943:
114:is a 3-manifold if it is a
10:
6157:
5888:Princeton University Press
5226:10.1051/0004-6361:20078777
5196:Astronomy and Astrophysics
5114:, article at PhysicsWorld.
5001:Swarup, G. Ananda (1974).
4976:Lubotzky–Sarnak conjecture
4898:Clay Mathematics Institute
4845:
4658:virtually Haken conjecture
4610:virtually Haken conjecture
4563:Virtually Haken conjecture
4556:
4489:, which states that every
4467:
4410:
4369:
4360:Ahlfors measure conjecture
4308:
4271:
4251:Thurston-Jørgensen theorem
3868:
3779:
3733:
3653:
3589:
3515:
3492:Annulus and Torus theorems
3481:{\displaystyle S^{2}\to M}
3328:{\displaystyle \partial M}
3182:Christos Papakyriakopoulos
3161:
3097:
3077:piecewise-linear structure
3056:
2995:
2958:
2923:Virtually Haken conjecture
2884:
2824:
2602:
2513:
2451:
2416:curvature that define the
2352:
2209:
2205:
2125:
1947:
1929:
54:that locally looks like a
6081:10.1515/9783110894516.239
4873:geometrization conjecture
4593:3-manifold with infinite
4378:ending lamination theorem
4356:ending lamination theorem
3662:Lickorish–Wallace theorem
3656:Lickorish–Wallace theorem
3650:Lickorish–Wallace theorem
2703:Knot and link complements
2501:and colleagues, that the
2365:dodecahedral tessellation
1214:can be computed from the
1194:the invariants above for
989:. It turns out the group
250:and topological methods.
6131:Low-dimensional topology
5462:10.2140/pjm.1996.172.139
4518:elliptization conjecture
4323:with finitely generated
3158:Loop and Sphere theorems
2557:Poincaré homology sphere
2476:binary icosahedral group
2156:, and is a special case
1959:Stereographic projection
806:there is a canonical map
654:, respectively; that is,
328:, we have the following
207:low-dimensional topology
5378:10.1112/jlms/s2-7.2.246
5218:2008A&A...482..747L
4650:Institut Henri Poincaré
4525:hyperbolization theorem
4458:Ricci flow with surgery
4291:Mostow rigidity theorem
4274:Hyperbolization theorem
4188:This theorem is due to
3871:Hyperbolic Dehn surgery
3768:cannot be a nontrivial
3570:The acronym JSJ is for
3176:is a generalization of
3142:Kneser–Haken finiteness
3023:that makes each leaf a
2539:. It is also known as
2122:Real projective 3-space
1940:over the real numbers.
1541:Moreover, a 3-manifold
34:An image from inside a
5853:Rolfsen, Dale (1976),
5529:10.3934/era.2009.16.44
5271:10.4310/jdg/1214441650
5019:10.1112/jlms/s2-8.1.13
4966:
4943:
4877:hyperbolic 3-manifolds
4828:
4827:{\displaystyle f|_{a}}
4796:
4770:
4699:
4669:Simple loop conjecture
4483:uniformization theorem
4390:hyperbolic 3-manifolds
4358:. It also implies the
4243:
4179:
4152:
4110:
4046:
3970:
3940:
3788:cyclic surgery theorem
3782:Cyclic surgery theorem
3776:Cyclic surgery theorem
3701:
3604:of 3-manifolds due to
3482:
3446:
3410:
3370:
3329:
3306:
3265:
2912:incompressible surface
2838:given by a hyperplane
2750:
2749:are a hyperbolic link.
2662:
2661:{\displaystyle \pi /3}
2470:itself) with a finite
2392:Hyperbolic space is a
2389:
2284:
2112:
2071:
2034:
2007:
1970:
1915:
1908:
1778:
1771:
1690:
1641:
1598:
1569:Second homotopy groups
1555:
1539:
1532:
1235:
1208:
1188:
1136:
1084:
1060:
1003:
983:
928:
889:
845:
838:
800:
793:
648:
628:
621:
314:
272:
222:incompressible surface
175:geometric group theory
135:
121:and if every point in
108:
39:
6037:10.1073/pnas.43.1.169
5786:Hempel, John (2004),
5713:Annals of Mathematics
5417:10.1112/blms/22.5.495
5046:Mathematische Annalen
4967:
4944:
4942:{\displaystyle (p,q)}
4908:Important conjectures
4890:Annals of Mathematics
4829:
4797:
4771:
4700:
4335:to the interior of a
4321:hyperbolic 3-manifold
4244:
4180:
4178:{\displaystyle u_{i}}
4153:
4111:
4047:
3971:
3969:{\displaystyle E_{i}}
3941:
3823:Seifert-fibered space
3702:
3700:{\displaystyle \pm 1}
3483:
3447:
3411:
3380:3-manifold such that
3371:
3330:
3312:not nullhomotopic in
3307:
3266:
2773:of constant negative
2745:
2693:Hyperbolic 3-manifold
2663:
2561:real projective space
2537:hyperbolic 3-manifold
2503:shape of the universe
2499:Observatoire de Paris
2404:. It is the model of
2362:
2285:
2128:Real projective space
2113:
2072:
2035:
2033:{\displaystyle S^{3}}
2008:
1957:
1909:
1782:
1772:
1691:
1649:
1642:
1599:
1556:
1533:
1243:
1236:
1234:{\displaystyle M_{i}}
1209:
1189:
1137:
1085:
1061:
1004:
984:
929:
890:
839:
808:
794:
656:
649:
622:
334:
315:
273:
136:
109:
64:shape of the universe
33:
6110:, Cornell University
6075:, DE GRUYTER, 2005,
5880:Thurston, William P.
5125:Luminet, Jean-Pierre
4953:
4921:
4894:Clay Research Awards
4858:Friedhelm Waldhausen
4806:
4780:
4709:
4677:
4581:, states that every
4485:for two-dimensional
4226:
4162:
4120:
4064:
3988:
3953:
3882:
3810:whose boundary is a
3716:Friedhelm Waldhausen
3688:
3641:finitely presentable
3526:, also known as the
3459:
3420:
3384:
3360:
3316:
3278:
3198:
3042:Foundational results
2942:essential lamination
2936:Essential lamination
2875:extended phase space
2769:that has a complete
2722:Spherical 3-manifold
2713:Seifert fiber spaces
2644:
2464:spherical 3-manifold
2223:
2081:
2044:
2017:
2006:{\displaystyle \pi }
1997:
1787:
1700:
1654:
1608:
1577:
1545:
1248:
1218:
1198:
1146:
1106:
1074:
1013:
1002:{\displaystyle \pi }
993:
938:
899:
851:
813:
661:
647:{\displaystyle \pi }
638:
339:
282:
278:be a 3-manifold and
262:
125:
98:
6028:1957PNAS...43..169P
5913:Adams, Colin Conrad
5399:Rubinstein, J. Hyam
5248:Epstein, David B.A.
5165:10.1038/nature01944
5157:2003Natur.425..593L
5087:Zimmermann, Bruno.
4514:Poincaré conjecture
4450:Richard S. Hamilton
4419:Poincaré conjecture
4413:Poincaré conjecture
4407:Poincaré conjecture
4311:Tameness conjecture
4099:
4081:
3976:is avoided for the
3790:states that, for a
3528:toral decomposition
2903:3-manifold that is
2867:classical mechanics
2863:symplectic geometry
2846:and specified by a
2779:hyperbolic geometry
2638:David B. A. Epstein
2526:Seifert–Weber space
2516:Seifert–Weber space
2510:Seifert–Weber space
2495:Jean-Pierre Luminet
2480:Poincaré conjecture
2406:hyperbolic geometry
1102:of two 3-manifolds
230:Heegaard splittings
179:hyperbolic geometry
6136:Geometric topology
6114:Strickland, Neil,
5790:, Providence, RI:
5704:Markovic, Vladimir
5549:Haglund and Wise,
5338:Johannson, Klaus,
5058:10.1007/BF01431437
4965:{\displaystyle pq}
4962:
4939:
4913:Cabling conjecture
4824:
4792:
4766:
4695:
4612:states that every
4479:topological spaces
4397:fundamental groups
4394:finitely generated
4329:topologically tame
4239:
4175:
4148:
4106:
4085:
4067:
4042:
3980:-th cusp for each
3966:
3948:exceptional slopes
3936:
3697:
3617:finitely generated
3602:fundamental groups
3598:Scott core theorem
3592:Scott core theorem
3586:Scott core theorem
3478:
3442:
3406:
3366:
3325:
3302:
3261:
3116:3-manifold is the
3108:states that every
3088:Heegaard splitting
3065:geometric topology
2967:Heegaard splitting
2961:Heegaard splitting
2955:Heegaard splitting
2905:sufficiently large
2751:
2658:
2615:Gieseking manifold
2605:Gieseking manifold
2599:Gieseking manifold
2593:hyperbolic 3-space
2418:Euclidean geometry
2390:
2381:cubic tessellation
2355:hyperbolic 3-space
2349:Hyperbolic 3-space
2280:
2108:
2067:
2030:
2003:
1971:
1904:
1767:
1686:
1637:
1594:
1551:
1528:
1526:
1231:
1204:
1184:
1132:
1080:
1056:
999:
979:
934:we get an element
924:
885:
834:
789:
787:
644:
634:and cohomology of
617:
615:
310:
268:
244:fundamental groups
224:and the theory of
211:geometric topology
187:TeichmĂĽller theory
131:
104:
40:
27:Mathematical space
6090:978-3-11-089451-6
5886:, Princeton, NJ:
5555:Hruska and Wise,
5368:, Second Series,
5141:(6958): 593–595.
4900:at a ceremony in
4885:Vladimir Markovic
4868:'s problem list.
4862:fundamental group
4626:fundamental group
4595:fundamental group
4331:, in other words
4325:fundamental group
3524:JSJ decomposition
3518:JSJ decomposition
3512:JSJ decomposition
3369:{\displaystyle M}
3344:Papakyriakopoulos
3129:prime 3-manifolds
3021:Riemannian metric
2852:Frobenius theorem
2797:Figure eight knot
2771:Riemannian metric
2619:cusped hyperbolic
2472:fundamental group
2422:elliptic geometry
2394:homogeneous space
2138:topological space
2056:
2013:acting freely on
1932:Euclidean 3-space
1926:Euclidean 3-space
1902:
1554:{\displaystyle M}
1207:{\displaystyle M}
1083:{\displaystyle M}
766:
535:
271:{\displaystyle M}
161:The topological,
151:Euclidean 3-space
134:{\displaystyle M}
107:{\displaystyle M}
93:topological space
52:topological space
16:(Redirected from
6148:
6111:
6104:Hatcher, Allen,
6093:
6067:
6057:
6039:
6001:
5991:
5966:
5937:
5908:
5875:
5849:
5825:Jaco, William H.
5820:
5800:10.1090/chel/349
5773:
5772:
5770:
5768:
5759:. Archived from
5753:
5747:
5746:
5729:
5720:(3): 1127–1190,
5696:
5690:
5689:
5687:
5675:
5669:
5658:
5652:
5651:
5649:
5637:
5631:
5630:
5628:
5616:
5610:
5609:
5607:
5595:
5589:
5581:Daniel T. Wise,
5579:
5570:
5547:
5541:
5540:
5531:
5504:
5498:
5497:
5495:
5483:
5474:
5473:
5464:
5434:
5428:
5427:
5395:
5389:
5388:
5356:
5350:
5336:
5330:
5323:
5317:
5316:
5314:
5290:
5284:
5283:
5273:
5244:
5238:
5237:
5211:
5191:
5185:
5184:
5150:
5148:astro-ph/0310253
5121:
5115:
5109:
5103:
5102:
5100:
5084:
5078:
5077:
5037:
5031:
5030:
4998:
4971:
4969:
4968:
4963:
4948:
4946:
4945:
4940:
4833:
4831:
4830:
4825:
4823:
4822:
4817:
4801:
4799:
4798:
4793:
4775:
4773:
4772:
4767:
4756:
4755:
4734:
4733:
4721:
4720:
4704:
4702:
4701:
4696:
4579:William Thurston
4571:, formulated by
4535:Grigori Perelman
4491:simply connected
4442:Grigori Perelman
4435:analogous result
4317:tameness theorem
4248:
4246:
4245:
4240:
4238:
4237:
4190:William Thurston
4184:
4182:
4181:
4176:
4174:
4173:
4157:
4155:
4154:
4149:
4147:
4146:
4137:
4132:
4131:
4115:
4113:
4112:
4107:
4098:
4093:
4080:
4075:
4051:
4049:
4048:
4043:
4038:
4037:
4019:
4018:
4006:
4005:
3984:. In addition,
3975:
3973:
3972:
3967:
3965:
3964:
3945:
3943:
3942:
3937:
3932:
3931:
3913:
3912:
3900:
3899:
3833:such that their
3742:Smith conjecture
3736:Smith conjecture
3730:Smith conjecture
3706:
3704:
3703:
3698:
3664:states that any
3631:, such that its
3487:
3485:
3484:
3479:
3471:
3470:
3451:
3449:
3448:
3443:
3432:
3431:
3415:
3413:
3412:
3407:
3396:
3395:
3375:
3373:
3372:
3367:
3334:
3332:
3331:
3326:
3311:
3309:
3308:
3303:
3301:
3300:
3288:
3270:
3268:
3267:
3262:
3235:
3234:
3219:
3218:
3127:) collection of
3081:smooth structure
3032:William Thurston
2836:smooth manifolds
2832:Contact geometry
2827:Contact geometry
2821:Contact geometry
2688:Homology spheres
2667:
2665:
2664:
2659:
2654:
2573:hyperbolic space
2420:, and models of
2410:Euclidean spaces
2289:
2287:
2286:
2281:
2276:
2275:
2263:
2262:
2250:
2249:
2237:
2236:
2231:
2117:
2115:
2114:
2109:
2104:
2099:
2098:
2076:
2074:
2073:
2068:
2057:
2054:
2039:
2037:
2036:
2031:
2029:
2028:
2012:
2010:
2009:
2004:
1913:
1911:
1910:
1905:
1903:
1901:
1897:
1896:
1878:
1877:
1864:
1860:
1859:
1841:
1840:
1819:
1813:
1799:
1798:
1776:
1774:
1773:
1768:
1757:
1756:
1741:
1740:
1725:
1724:
1712:
1711:
1695:
1693:
1692:
1687:
1679:
1678:
1666:
1665:
1646:
1644:
1643:
1638:
1633:
1632:
1620:
1619:
1603:
1601:
1600:
1595:
1584:
1560:
1558:
1557:
1552:
1537:
1535:
1534:
1529:
1527:
1520:
1519:
1507:
1506:
1485:
1484:
1472:
1471:
1446:
1445:
1429:
1428:
1416:
1415:
1394:
1393:
1381:
1380:
1355:
1354:
1338:
1337:
1325:
1324:
1303:
1302:
1290:
1289:
1264:
1263:
1240:
1238:
1237:
1232:
1230:
1229:
1213:
1211:
1210:
1205:
1193:
1191:
1190:
1185:
1183:
1182:
1164:
1163:
1141:
1139:
1138:
1133:
1131:
1130:
1118:
1117:
1089:
1087:
1086:
1081:
1065:
1063:
1062:
1057:
1052:
1038:
1037:
1025:
1024:
1008:
1006:
1005:
1000:
988:
986:
985:
980:
963:
962:
950:
949:
933:
931:
930:
925:
911:
910:
894:
892:
891:
886:
875:
874:
843:
841:
840:
835:
798:
796:
795:
790:
788:
781:
767:
764:
752:
738:
737:
709:
691:
677:
676:
653:
651:
650:
645:
626:
624:
623:
618:
616:
612:
593:
592:
567:
566:
550:
536:
533:
517:
516:
491:
490:
462:
440:
439:
414:
413:
400:
381:
380:
355:
354:
326:Hurewicz theorem
322:Poincare duality
319:
317:
316:
311:
300:
299:
277:
275:
274:
269:
169:fields, such as
163:piecewise-linear
140:
138:
137:
132:
116:second-countable
113:
111:
110:
105:
21:
6156:
6155:
6151:
6150:
6149:
6147:
6146:
6145:
6121:
6120:
6100:
6091:
6070:
5956:
5927:
5898:
5865:
5855:Knots and Links
5839:
5810:
5782:
5780:Further reading
5777:
5776:
5766:
5764:
5763:on June 4, 2012
5755:
5754:
5750:
5697:
5693:
5676:
5672:
5659:
5655:
5638:
5634:
5617:
5613:
5596:
5592:
5580:
5573:
5566:
5560:
5554:
5548:
5544:
5508:Wise, Daniel T.
5505:
5501:
5484:
5477:
5439:Scott, G. Peter
5435:
5431:
5396:
5392:
5360:Scott, G. Peter
5357:
5353:
5337:
5333:
5324:
5320:
5291:
5287:
5245:
5241:
5192:
5188:
5122:
5118:
5110:
5106:
5085:
5081:
5038:
5034:
4999:
4995:
4990:
4978:
4954:
4951:
4950:
4922:
4919:
4918:
4915:
4910:
4850:
4844:
4818:
4813:
4812:
4807:
4804:
4803:
4781:
4778:
4777:
4751:
4747:
4729:
4725:
4716:
4712:
4710:
4707:
4706:
4678:
4675:
4674:
4671:
4630:virtually Haken
4565:
4557:Main articles:
4555:
4529:Haken manifolds
4516:and Thurston's
4510:Thurston (1982)
4494:Riemann surface
4472:
4466:
4415:
4409:
4374:
4368:
4313:
4307:
4276:
4270:
4233:
4229:
4227:
4224:
4223:
4169:
4165:
4163:
4160:
4159:
4142:
4138:
4133:
4127:
4123:
4121:
4118:
4117:
4094:
4089:
4076:
4071:
4065:
4062:
4061:
4033:
4029:
4014:
4010:
4001:
3997:
3989:
3986:
3985:
3960:
3956:
3954:
3951:
3950:
3927:
3923:
3908:
3904:
3895:
3891:
3883:
3880:
3879:
3873:
3867:
3806:three-manifold
3784:
3778:
3762:fixed point set
3738:
3732:
3724:
3713:
3689:
3686:
3685:
3658:
3652:
3594:
3588:
3580:Klaus Johannson
3564:Seifert-fibered
3520:
3514:
3498:annulus theorem
3494:
3466:
3462:
3460:
3457:
3456:
3427:
3423:
3421:
3418:
3417:
3391:
3387:
3385:
3382:
3381:
3361:
3358:
3357:
3317:
3314:
3313:
3296:
3292:
3284:
3279:
3276:
3275:
3230:
3226:
3214:
3210:
3199:
3196:
3195:
3170:
3162:Main articles:
3160:
3144:
3102:
3096:
3069:Moise's theorem
3061:
3059:Moise's theorem
3055:
3053:Moise's theorem
3044:
3025:minimal surface
3017:Dennis Sullivan
3000:
2994:
2963:
2957:
2938:
2919:virtually Haken
2889:
2883:
2829:
2823:
2818:
2807:Borromean rings
2783:hyperbolic knot
2755:hyperbolic link
2747:Borromean rings
2740:
2674:
2650:
2645:
2642:
2641:
2607:
2601:
2530:Herbert Seifert
2528:(introduced by
2518:
2512:
2456:
2450:
2436:covered by the
2434:amount of space
2374:
2357:
2351:
2343:complex numbers
2309:of the integer
2271:
2267:
2258:
2254:
2245:
2241:
2232:
2227:
2226:
2224:
2221:
2220:
2214:
2208:
2200:universal cover
2150:smooth manifold
2130:
2124:
2100:
2094:
2090:
2082:
2079:
2078:
2053:
2045:
2042:
2041:
2024:
2020:
2018:
2015:
2014:
1998:
1995:
1994:
1979:Euclidean space
1952:
1946:
1934:
1928:
1923:
1892:
1888:
1873:
1869:
1865:
1855:
1851:
1836:
1832:
1815:
1814:
1812:
1794:
1790:
1788:
1785:
1784:
1752:
1748:
1736:
1732:
1720:
1716:
1707:
1703:
1701:
1698:
1697:
1674:
1670:
1661:
1657:
1655:
1652:
1651:
1628:
1624:
1615:
1611:
1609:
1606:
1605:
1580:
1578:
1575:
1574:
1571:
1546:
1543:
1542:
1525:
1524:
1515:
1511:
1502:
1498:
1480:
1476:
1467:
1463:
1456:
1441:
1437:
1434:
1433:
1424:
1420:
1411:
1407:
1389:
1385:
1376:
1372:
1365:
1350:
1346:
1343:
1342:
1333:
1329:
1320:
1316:
1298:
1294:
1285:
1281:
1274:
1259:
1255:
1251:
1249:
1246:
1245:
1241:. In particular
1225:
1221:
1219:
1216:
1215:
1199:
1196:
1195:
1178:
1174:
1159:
1155:
1147:
1144:
1143:
1126:
1122:
1113:
1109:
1107:
1104:
1103:
1096:
1075:
1072:
1071:
1048:
1033:
1029:
1020:
1016:
1014:
1011:
1010:
994:
991:
990:
958:
954:
945:
941:
939:
936:
935:
906:
902:
900:
897:
896:
870:
866:
852:
849:
848:
814:
811:
810:
804:Postnikov tower
786:
785:
777:
763:
756:
748:
733:
729:
726:
725:
705:
695:
687:
672:
668:
664:
662:
659:
658:
639:
636:
635:
614:
613:
608:
606:
588:
584:
577:
562:
558:
555:
554:
546:
532:
530:
512:
508:
501:
486:
482:
479:
478:
458:
453:
435:
431:
424:
409:
405:
402:
401:
396:
394:
376:
372:
365:
350:
346:
342:
340:
337:
336:
330:homology groups
295:
291:
283:
280:
279:
263:
260:
259:
256:
226:Haken manifolds
159:
126:
123:
122:
119:Hausdorff space
99:
96:
95:
89:
84:
28:
23:
22:
15:
12:
11:
5:
6154:
6144:
6143:
6138:
6133:
6119:
6118:
6112:
6099:
6098:External links
6096:
6095:
6094:
6089:
6068:
6022:(1): 169–172.
6008:(1957-01-15).
6002:
5982:(3): 357–382.
5967:
5954:
5938:
5925:
5909:
5896:
5876:
5863:
5850:
5837:
5821:
5808:
5781:
5778:
5775:
5774:
5748:
5691:
5670:
5653:
5632:
5611:
5590:
5571:
5561:Hsu and Wise,
5542:
5510:(2009-10-29),
5499:
5475:
5455:(1): 139–150,
5437:Harris, Luke;
5429:
5411:(5): 495–498,
5390:
5372:(2): 246–250,
5351:
5331:
5318:
5285:
5239:
5202:(3): 747–753.
5186:
5129:Weeks, Jeffrey
5116:
5104:
5098:10.1.1.218.102
5079:
5032:
4992:
4991:
4989:
4986:
4977:
4974:
4961:
4958:
4938:
4935:
4932:
4929:
4926:
4914:
4911:
4909:
4906:
4846:Main article:
4843:
4840:
4821:
4816:
4811:
4791:
4788:
4785:
4765:
4762:
4759:
4754:
4750:
4746:
4743:
4740:
4737:
4732:
4728:
4724:
4719:
4715:
4694:
4691:
4688:
4685:
4682:
4670:
4667:
4638:Haken manifold
4634:covering space
4554:
4551:
4468:Main article:
4465:
4462:
4423:Henri Poincaré
4411:Main article:
4408:
4405:
4386:Richard Canary
4370:Main article:
4367:
4364:
4344:Danny Calegari
4309:Main article:
4306:
4303:
4272:Main article:
4269:
4266:
4262:Weeks manifold
4236:
4232:
4172:
4168:
4145:
4141:
4136:
4130:
4126:
4105:
4102:
4097:
4092:
4088:
4084:
4079:
4074:
4070:
4041:
4036:
4032:
4028:
4025:
4022:
4017:
4013:
4009:
4004:
4000:
3996:
3993:
3963:
3959:
3935:
3930:
3926:
3922:
3919:
3916:
3911:
3907:
3903:
3898:
3894:
3890:
3887:
3869:Main article:
3866:
3863:
3829:are slopes on
3780:Main article:
3777:
3774:
3750:diffeomorphism
3734:Main article:
3731:
3728:
3723:
3720:
3712:
3709:
3696:
3693:
3654:Main article:
3651:
3648:
3606:G. Peter Scott
3590:Main article:
3587:
3584:
3568:
3567:
3553:incompressible
3516:Main article:
3513:
3510:
3493:
3490:
3477:
3474:
3469:
3465:
3441:
3438:
3435:
3430:
3426:
3405:
3402:
3399:
3394:
3390:
3365:
3340:sphere theorem
3324:
3321:
3299:
3295:
3291:
3287:
3283:
3272:
3271:
3259:
3256:
3253:
3250:
3247:
3244:
3241:
3238:
3233:
3229:
3225:
3222:
3217:
3213:
3209:
3206:
3203:
3186:Sphere theorem
3159:
3156:
3143:
3140:
3134:A manifold is
3098:Main article:
3095:
3092:
3073:Edwin E. Moise
3057:Main article:
3054:
3051:
3043:
3040:
3004:taut foliation
2998:Taut foliation
2996:Main article:
2993:
2992:Taut foliation
2990:
2976:Every closed,
2959:Main article:
2956:
2953:
2937:
2934:
2901:P²-irreducible
2893:Haken manifold
2887:Haken manifold
2885:Main article:
2882:
2881:Haken manifold
2879:
2844:tangent bundle
2825:Main article:
2822:
2819:
2817:
2814:
2810:
2809:
2804:
2802:Whitehead link
2799:
2739:
2736:
2735:
2734:
2729:
2724:
2719:
2717:Circle bundles
2710:
2705:
2700:
2695:
2690:
2685:
2683:Haken manifold
2680:
2678:Graph manifold
2673:
2670:
2657:
2653:
2649:
2623:non-orientable
2603:Main article:
2600:
2597:
2578:quotient space
2569:dihedral angle
2514:Main article:
2511:
2508:
2452:Main article:
2449:
2446:
2353:Main article:
2350:
2347:
2291:
2290:
2279:
2274:
2270:
2266:
2261:
2257:
2253:
2248:
2244:
2240:
2235:
2230:
2210:Main article:
2207:
2204:
2126:Main article:
2123:
2120:
2107:
2103:
2097:
2093:
2089:
2086:
2066:
2063:
2060:
2052:
2049:
2027:
2023:
2002:
1948:Main article:
1945:
1942:
1930:Main article:
1927:
1924:
1922:
1919:
1900:
1895:
1891:
1887:
1884:
1881:
1876:
1872:
1868:
1863:
1858:
1854:
1850:
1847:
1844:
1839:
1835:
1831:
1828:
1825:
1822:
1818:
1811:
1808:
1805:
1802:
1797:
1793:
1766:
1763:
1760:
1755:
1751:
1747:
1744:
1739:
1735:
1731:
1728:
1723:
1719:
1715:
1710:
1706:
1685:
1682:
1677:
1673:
1669:
1664:
1660:
1636:
1631:
1627:
1623:
1618:
1614:
1593:
1590:
1587:
1583:
1570:
1567:
1550:
1523:
1518:
1514:
1510:
1505:
1501:
1497:
1494:
1491:
1488:
1483:
1479:
1475:
1470:
1466:
1462:
1459:
1457:
1455:
1452:
1449:
1444:
1440:
1436:
1435:
1432:
1427:
1423:
1419:
1414:
1410:
1406:
1403:
1400:
1397:
1392:
1388:
1384:
1379:
1375:
1371:
1368:
1366:
1364:
1361:
1358:
1353:
1349:
1345:
1344:
1341:
1336:
1332:
1328:
1323:
1319:
1315:
1312:
1309:
1306:
1301:
1297:
1293:
1288:
1284:
1280:
1277:
1275:
1273:
1270:
1267:
1262:
1258:
1254:
1253:
1228:
1224:
1203:
1181:
1177:
1173:
1170:
1167:
1162:
1158:
1154:
1151:
1129:
1125:
1121:
1116:
1112:
1095:
1094:Connected sums
1092:
1079:
1055:
1051:
1047:
1044:
1041:
1036:
1032:
1028:
1023:
1019:
998:
978:
975:
972:
969:
966:
961:
957:
953:
948:
944:
923:
920:
917:
914:
909:
905:
884:
881:
878:
873:
869:
865:
862:
859:
856:
833:
830:
827:
824:
821:
818:
784:
780:
776:
773:
770:
762:
759:
757:
755:
751:
747:
744:
741:
736:
732:
728:
727:
724:
721:
718:
715:
712:
708:
704:
701:
698:
696:
694:
690:
686:
683:
680:
675:
671:
667:
666:
643:
632:group homology
611:
607:
605:
602:
599:
596:
591:
587:
583:
580:
578:
576:
573:
570:
565:
561:
557:
556:
553:
549:
545:
542:
539:
531:
529:
526:
523:
520:
515:
511:
507:
504:
502:
500:
497:
494:
489:
485:
481:
480:
477:
474:
471:
468:
465:
461:
457:
454:
452:
449:
446:
443:
438:
434:
430:
427:
425:
423:
420:
417:
412:
408:
404:
403:
399:
395:
393:
390:
387:
384:
379:
375:
371:
368:
366:
364:
361:
358:
353:
349:
345:
344:
309:
306:
303:
298:
294:
290:
287:
267:
255:
252:
199:Floer homology
158:
155:
130:
103:
88:
85:
83:
80:
26:
9:
6:
4:
3:
2:
6153:
6142:
6139:
6137:
6134:
6132:
6129:
6128:
6126:
6117:
6113:
6109:
6108:
6102:
6101:
6092:
6086:
6082:
6078:
6074:
6069:
6065:
6061:
6056:
6051:
6047:
6043:
6038:
6033:
6029:
6025:
6021:
6017:
6016:
6011:
6007:
6003:
5999:
5995:
5990:
5985:
5981:
5977:
5973:
5968:
5965:
5961:
5957:
5955:0-8218-1040-5
5951:
5947:
5943:
5939:
5936:
5932:
5928:
5926:0-8050-7380-9
5922:
5918:
5914:
5910:
5907:
5903:
5899:
5897:0-691-08304-5
5893:
5889:
5885:
5881:
5877:
5874:
5870:
5866:
5864:0-914098-16-0
5860:
5856:
5851:
5848:
5844:
5840:
5838:0-8218-1693-4
5834:
5830:
5826:
5822:
5819:
5815:
5811:
5809:0-8218-3695-1
5805:
5801:
5797:
5793:
5789:
5784:
5783:
5762:
5758:
5752:
5745:
5741:
5737:
5733:
5728:
5723:
5719:
5715:
5714:
5709:
5705:
5701:
5695:
5686:
5681:
5674:
5668:
5667:
5662:
5657:
5648:
5643:
5636:
5627:
5622:
5615:
5606:
5601:
5594:
5588:
5584:
5578:
5576:
5569:
5564:
5558:
5552:
5546:
5539:
5535:
5530:
5525:
5521:
5517:
5513:
5509:
5503:
5494:
5489:
5482:
5480:
5472:
5468:
5463:
5458:
5454:
5450:
5449:
5444:
5440:
5433:
5426:
5422:
5418:
5414:
5410:
5406:
5405:
5400:
5394:
5387:
5383:
5379:
5375:
5371:
5367:
5366:
5361:
5355:
5349:
5348:3-540-09714-7
5345:
5341:
5335:
5328:
5322:
5313:
5308:
5304:
5300:
5296:
5289:
5281:
5277:
5272:
5267:
5263:
5259:
5258:
5253:
5249:
5243:
5235:
5231:
5227:
5223:
5219:
5215:
5210:
5205:
5201:
5197:
5190:
5182:
5178:
5174:
5170:
5166:
5162:
5158:
5154:
5149:
5144:
5140:
5136:
5135:
5130:
5126:
5120:
5113:
5108:
5099:
5094:
5090:
5083:
5075:
5071:
5067:
5063:
5059:
5055:
5052:(2): 89–102.
5051:
5047:
5043:
5036:
5028:
5024:
5020:
5016:
5012:
5008:
5004:
4997:
4993:
4985:
4983:
4973:
4959:
4956:
4933:
4930:
4927:
4905:
4903:
4899:
4895:
4891:
4886:
4882:
4878:
4874:
4871:Assuming the
4869:
4867:
4863:
4859:
4855:
4849:
4839:
4837:
4819:
4809:
4789:
4786:
4783:
4760:
4752:
4748:
4738:
4730:
4726:
4722:
4717:
4713:
4692:
4686:
4683:
4680:
4666:
4663:
4659:
4655:
4651:
4646:
4641:
4639:
4635:
4631:
4627:
4623:
4619:
4615:
4611:
4606:
4604:
4600:
4597:has a finite
4596:
4592:
4588:
4584:
4580:
4577:
4576:mathematician
4574:
4570:
4564:
4560:
4550:
4548:
4544:
4540:
4536:
4532:
4530:
4527:implies that
4526:
4521:
4519:
4515:
4511:
4507:
4503:
4499:
4495:
4492:
4488:
4484:
4480:
4476:
4471:
4461:
4459:
4455:
4451:
4447:
4443:
4438:
4436:
4432:
4428:
4424:
4420:
4414:
4404:
4402:
4398:
4395:
4391:
4387:
4383:
4382:Jeffrey Brock
4379:
4373:
4363:
4361:
4357:
4353:
4349:
4345:
4340:
4338:
4334:
4330:
4326:
4322:
4318:
4312:
4302:
4299:
4294:
4292:
4287:
4285:
4281:
4275:
4265:
4263:
4258:
4256:
4252:
4234:
4230:
4222:
4218:
4214:
4210:
4205:
4203:
4197:
4195:
4191:
4186:
4170:
4166:
4143:
4139:
4134:
4128:
4124:
4095:
4090:
4086:
4082:
4077:
4072:
4068:
4059:
4055:
4052:converges to
4034:
4030:
4026:
4023:
4020:
4015:
4011:
4007:
4002:
3998:
3991:
3983:
3979:
3961:
3957:
3949:
3928:
3924:
3920:
3917:
3914:
3909:
3905:
3901:
3896:
3892:
3885:
3877:
3872:
3862:
3860:
3856:
3852:
3849:representing
3848:
3844:
3840:
3836:
3835:Dehn fillings
3832:
3828:
3824:
3820:
3816:
3813:
3809:
3805:
3801:
3797:
3793:
3789:
3783:
3773:
3771:
3767:
3763:
3759:
3755:
3751:
3747:
3743:
3737:
3727:
3719:
3717:
3708:
3694:
3691:
3683:
3679:
3675:
3671:
3667:
3663:
3657:
3647:
3644:
3642:
3638:
3634:
3633:inclusion map
3630:
3626:
3623:, called the
3622:
3618:
3614:
3609:
3607:
3603:
3599:
3593:
3583:
3581:
3577:
3573:
3565:
3561:
3557:
3554:
3551:
3547:
3543:
3540:
3537:
3536:
3535:
3533:
3529:
3525:
3519:
3509:
3507:
3506:torus theorem
3502:
3499:
3489:
3475:
3467:
3463:
3455:
3436:
3428:
3424:
3400:
3392:
3388:
3379:
3363:
3354:
3351:
3349:
3345:
3341:
3336:
3322:
3297:
3293:
3281:
3254:
3248:
3245:
3231:
3227:
3220:
3215:
3211:
3204:
3201:
3194:
3193:
3192:
3189:
3187:
3183:
3179:
3175:
3169:
3165:
3155:
3153:
3149:
3139:
3137:
3132:
3130:
3126:
3125:homeomorphism
3123:
3120:of a unique (
3119:
3118:connected sum
3115:
3111:
3107:
3101:
3091:
3089:
3084:
3082:
3078:
3074:
3070:
3066:
3060:
3050:
3047:
3039:
3037:
3033:
3028:
3026:
3022:
3018:
3013:
3009:
3005:
2999:
2989:
2987:
2983:
2979:
2974:
2972:
2968:
2962:
2952:
2949:
2947:
2943:
2933:
2931:
2926:
2924:
2920:
2915:
2913:
2910:
2906:
2902:
2898:
2894:
2888:
2878:
2876:
2872:
2868:
2864:
2859:
2857:
2853:
2849:
2845:
2841:
2837:
2833:
2828:
2813:
2808:
2805:
2803:
2800:
2798:
2795:
2794:
2793:
2790:
2788:
2784:
2780:
2777:, i.e. has a
2776:
2772:
2768:
2764:
2760:
2756:
2748:
2744:
2733:
2730:
2728:
2725:
2723:
2720:
2718:
2714:
2711:
2709:
2706:
2704:
2701:
2699:
2696:
2694:
2691:
2689:
2686:
2684:
2681:
2679:
2676:
2675:
2669:
2655:
2651:
2647:
2639:
2635:
2630:
2628:
2624:
2620:
2616:
2612:
2606:
2596:
2594:
2590:
2587:
2583:
2579:
2574:
2570:
2564:
2562:
2558:
2553:
2548:
2546:
2542:
2538:
2535:
2531:
2527:
2523:
2517:
2507:
2504:
2500:
2496:
2492:
2488:
2483:
2481:
2477:
2473:
2469:
2465:
2461:
2455:
2445:
2443:
2442:exponentially
2439:
2435:
2431:
2427:
2423:
2419:
2415:
2411:
2407:
2403:
2399:
2395:
2388:
2387:
2382:
2378:
2372:
2371:
2366:
2361:
2356:
2346:
2344:
2340:
2336:
2333:
2329:
2326:
2321:
2319:
2315:
2312:
2308:
2304:
2300:
2296:
2293:The 3-torus,
2277:
2272:
2268:
2264:
2259:
2255:
2251:
2246:
2242:
2238:
2233:
2219:
2218:
2217:
2213:
2203:
2201:
2197:
2193:
2189:
2185:
2181:
2177:
2176:diffeomorphic
2173:
2169:
2167:
2163:
2159:
2155:
2152:of dimension
2151:
2147:
2143:
2139:
2135:
2129:
2119:
2105:
2101:
2095:
2091:
2087:
2084:
2061:
2047:
2025:
2021:
2000:
1992:
1988:
1984:
1980:
1976:
1968:
1964:
1960:
1956:
1951:
1941:
1939:
1933:
1918:
1914:
1893:
1889:
1885:
1882:
1879:
1874:
1870:
1856:
1852:
1848:
1845:
1842:
1837:
1833:
1823:
1809:
1803:
1795:
1791:
1781:
1777:
1764:
1761:
1753:
1749:
1742:
1737:
1733:
1729:
1721:
1717:
1708:
1704:
1683:
1675:
1671:
1667:
1662:
1658:
1648:
1629:
1625:
1616:
1612:
1588:
1566:
1564:
1548:
1538:
1516:
1512:
1503:
1499:
1495:
1492:
1489:
1481:
1477:
1468:
1464:
1460:
1458:
1450:
1442:
1438:
1425:
1421:
1412:
1408:
1404:
1401:
1398:
1390:
1386:
1377:
1373:
1369:
1367:
1359:
1351:
1347:
1334:
1330:
1321:
1317:
1313:
1310:
1307:
1299:
1295:
1286:
1282:
1278:
1276:
1268:
1260:
1256:
1242:
1226:
1222:
1201:
1179:
1175:
1168:
1160:
1156:
1152:
1149:
1127:
1123:
1114:
1110:
1101:
1100:connected sum
1091:
1077:
1069:
1068:homotopy type
1045:
1042:
1034:
1030:
1026:
1021:
1017:
996:
970:
959:
955:
951:
946:
942:
918:
915:
907:
903:
879:
871:
867:
863:
857:
844:
831:
828:
822:
819:
816:
807:
805:
799:
774:
771:
760:
758:
745:
742:
734:
730:
719:
716:
713:
706:
702:
699:
697:
684:
681:
673:
669:
655:
641:
633:
627:
603:
597:
589:
585:
581:
579:
571:
563:
559:
543:
540:
527:
521:
513:
509:
505:
503:
495:
487:
483:
472:
469:
466:
459:
455:
450:
444:
436:
432:
428:
426:
418:
410:
406:
391:
385:
377:
373:
369:
367:
359:
351:
347:
333:
331:
327:
323:
304:
296:
292:
288:
285:
265:
251:
249:
245:
240:
237:
233:
231:
227:
223:
219:
214:
212:
208:
204:
200:
196:
192:
188:
184:
183:number theory
180:
176:
172:
166:
164:
154:
152:
148:
144:
143:neighbourhood
128:
120:
117:
101:
94:
79:
77:
76:tangent plane
73:
70:looks like a
69:
65:
61:
57:
53:
49:
45:
37:
32:
19:
6106:
6072:
6019:
6013:
5979:
5975:
5945:
5916:
5883:
5854:
5828:
5787:
5765:. Retrieved
5761:the original
5751:
5717:
5711:
5700:Kahn, Jeremy
5694:
5673:
5665:
5661:Robion Kirby
5656:
5635:
5614:
5593:
5582:
5562:
5556:
5550:
5545:
5519:
5515:
5502:
5452:
5446:
5432:
5408:
5402:
5393:
5369:
5363:
5354:
5339:
5334:
5326:
5321:
5302:
5298:
5288:
5264:(1): 67–80.
5261:
5255:
5242:
5199:
5195:
5189:
5138:
5132:
5119:
5107:
5088:
5082:
5049:
5045:
5035:
5013:(1): 13–21.
5010:
5006:
4996:
4981:
4979:
4916:
4870:
4866:Robion Kirby
4853:
4851:
4672:
4642:
4629:
4609:
4607:
4568:
4566:
4533:
4523:Thurston's
4522:
4474:
4473:
4457:
4439:
4416:
4377:
4375:
4341:
4339:3-manifold.
4333:homeomorphic
4316:
4314:
4297:
4295:
4288:
4283:
4279:
4277:
4259:
4250:
4221:ordinal type
4216:
4206:
4198:
4193:
4187:
4057:
4053:
3981:
3977:
3947:
3875:
3874:
3858:
3854:
3850:
3846:
3842:
3838:
3830:
3826:
3818:
3814:
3807:
3787:
3785:
3765:
3758:finite order
3745:
3741:
3739:
3725:
3714:
3674:Dehn surgery
3661:
3659:
3645:
3628:
3625:compact core
3624:
3610:
3597:
3595:
3576:Peter Shalen
3572:William Jaco
3569:
3527:
3523:
3521:
3505:
3503:
3497:
3495:
3355:
3352:
3339:
3337:
3273:
3190:
3178:Dehn's lemma
3174:loop theorem
3173:
3171:
3164:Loop theorem
3151:
3147:
3145:
3135:
3133:
3105:
3103:
3085:
3071:, proved by
3068:
3062:
3048:
3045:
3029:
3003:
3001:
2975:
2971:handlebodies
2966:
2964:
2950:
2941:
2939:
2929:
2927:
2918:
2916:
2904:
2892:
2890:
2860:
2840:distribution
2831:
2830:
2811:
2791:
2782:
2754:
2752:
2732:Torus bundle
2631:
2614:
2608:
2589:tessellation
2565:
2552:dodecahedron
2549:
2544:
2540:
2525:
2519:
2484:
2457:
2430:saddle point
2391:
2384:
2368:
2322:
2313:
2302:
2298:
2294:
2292:
2215:
2198:that is the
2187:
2183:
2171:
2170:
2166:Grassmannian
2161:
2157:
2153:
2141:
2133:
2131:
1972:
1938:vector space
1935:
1916:
1783:
1779:
1650:
1572:
1562:
1540:
1244:
1097:
846:
809:
801:
657:
629:
335:
257:
248:group theory
241:
234:
215:
195:gauge theory
167:
160:
147:homeomorphic
90:
66:. Just as a
47:
41:
6141:3-manifolds
5942:Bing, R. H.
5788:3-manifolds
4881:Jeremy Kahn
4836:David Gabai
4645:Daniel Wise
4622:irreducible
4601:which is a
4587:irreducible
4452:to use the
4401:laminations
4348:David Gabai
4202:Gromov norm
3804:irreducible
3760:, then the
3678:framed link
3637:isomorphism
3635:induces an
3621:submanifold
3539:Irreducible
3532:topological
3036:David Gabai
3008:codimension
2871:phase space
2634:tetrahedron
2611:mathematics
2522:mathematics
2377:dodecahedra
2339:unit circle
2305:modulo the
171:knot theory
44:mathematics
18:3-manifolds
6125:Categories
4988:References
4802:such that
4652:in Paris,
4618:orientable
4539:Ricci flow
4506:hyperbolic
4454:Ricci flow
4209:continuous
3800:orientable
3670:orientable
3629:Scott core
3542:orientable
3378:orientable
3114:orientable
2978:orientable
2946:lamination
2767:complement
2708:Lens space
2555:gives the
2424:(like the
2202:of SO(3).
2144:. It is a
2040:via a map
1991:dimensions
236:Thurston's
87:Definition
82:Principles
48:3-manifold
6046:0027-8424
5998:0273-0979
5727:0910.5501
5685:0910.5501
5647:1012.2828
5626:0910.5501
5605:1204.2810
5522:: 44–55,
5493:0908.3609
5209:0801.0006
5093:CiteSeerX
5074:120672504
5066:1432-1807
5027:1469-7750
4787:⊂
4784:α
4749:π
4745:→
4727:π
4723::
4718:⋆
4690:→
4684::
4591:atoroidal
4502:spherical
4498:Euclidean
4235:ω
4231:ω
4104:∞
4101:→
4024:…
3918:…
3878:states:
3821:is not a
3796:connected
3692:±
3560:atoroidal
3473:→
3454:embedding
3425:π
3389:π
3320:∂
3290:∂
3252:∂
3240:→
3224:∂
3205::
3012:foliation
2930:hierarchy
2909:two-sided
2856:foliation
2787:component
2775:curvature
2698:I-bundles
2648:π
2402:curvature
2400:negative
2335:Lie group
2265:×
2252:×
2196:Lie group
2136:, is the
2106:π
2051:→
2048:π
2001:π
1967:conformal
1963:meridians
1890:σ
1883:⋯
1871:σ
1853:σ
1846:…
1834:σ
1824:π
1792:π
1762:⊂
1743:−
1730:⊂
1705:σ
1681:→
1659:σ
1613:π
1589:π
1500:π
1496:∗
1493:⋯
1490:∗
1465:π
1439:π
1405:⊕
1402:⋯
1399:⊕
1314:⊕
1311:⋯
1308:⊕
1172:#
1169:⋯
1166:#
1120:#
1043:π
1027:∈
1018:ζ
997:π
960:∗
943:ζ
919:π
864:∈
832:π
826:→
772:π
761:≅
743:π
720:π
714:π
703:π
700:≅
682:π
642:π
541:π
473:π
467:π
456:π
293:π
286:π
6064:16589993
5944:(1983),
5915:(2004),
5882:(1997),
5827:(1980),
5744:32593851
5706:(2012),
5441:(1996),
5305:: 1–43.
5173:14534579
4654:Ian Agol
4573:American
4487:surfaces
4354:and the
4116:for all
3754:3-sphere
3682:3-sphere
3550:embedded
2848:one-form
2763:3-sphere
2468:3-sphere
2460:Poincaré
2426:3-sphere
2398:constant
2328:manifold
1950:3-sphere
1944:3-sphere
324:and the
218:surfaces
145:that is
60:manifold
6024:Bibcode
5964:0928227
5935:2079925
5906:1435975
5873:1277811
5847:0565450
5818:2098385
5767:Apr 30,
5538:2558631
5471:1379290
5425:1082023
5386:0326737
5280:0918457
5234:1616362
5214:Bibcode
5181:4380713
5153:Bibcode
4896:by the
4614:compact
4543:surgery
4337:compact
4060:as all
3792:compact
3752:of the
3680:in the
3615:) with
3613:compact
3546:isotopy
3530:, is a
3346: (
3110:compact
2921:. The
2897:compact
2842:in the
2761:in the
2586:regular
2580:of the
2497:of the
2332:abelian
2325:compact
2311:lattice
2206:3-torus
2192:Spin(3)
2168:space.
2164:) of a
2146:compact
1983:surface
36:3-torus
6087:
6062:
6055:528404
6052:
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5536:
5469:
5423:
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5346:
5278:
5232:
5179:
5171:
5134:Nature
5095:
5072:
5064:
5025:
4902:Oxford
4583:closed
4427:closed
4255:Gromov
4213:proper
3666:closed
3578:, and
3376:be an
2613:, the
2534:closed
2438:3-ball
2307:action
1975:sphere
1696:where
201:, and
141:has a
68:sphere
58:. A 3-
5740:S2CID
5722:arXiv
5680:arXiv
5642:arXiv
5621:arXiv
5600:arXiv
5488:arXiv
5230:S2CID
5204:arXiv
5177:S2CID
5143:arXiv
5070:S2CID
4599:cover
4541:with
4504:, or
4446:arXiv
4392:with
3817:, if
3812:torus
3748:is a
3684:with
3676:on a
3274:with
3136:prime
3122:up to
3006:is a
2986:Smale
2982:Moise
2944:is a
2895:is a
2765:with
2757:is a
2617:is a
2412:with
2375:Four
2194:is a
2180:SO(3)
2077:, so
1563:prime
895:into
72:plane
50:is a
6085:ISBN
6060:PMID
6042:ISSN
5994:ISSN
5950:ISBN
5921:ISBN
5892:ISBN
5859:ISBN
5833:ISBN
5804:ISBN
5769:2020
5344:ISBN
5169:PMID
5062:ISSN
5023:ISSN
5011:s2-8
4883:and
4852:The
4608:The
4567:The
4561:and
4431:loop
4376:The
4346:and
4315:The
4289:The
3853:and
3841:and
3825:and
3786:The
3770:knot
3740:The
3660:The
3596:The
3556:tori
3522:The
3504:The
3496:The
3356:Let
3348:1957
3338:The
3172:The
3166:and
3104:The
3079:and
3034:and
2781:. A
2759:link
2627:1912
2584:, a
2543:and
2491:WMAP
2458:The
2414:zero
2318:cube
2178:to)
2174:is (
2160:(1,
1987:ball
242:The
46:, a
6077:doi
6050:PMC
6032:doi
5984:doi
5796:doi
5732:doi
5718:175
5524:doi
5457:doi
5453:172
5413:doi
5374:doi
5307:doi
5303:217
5266:doi
5222:doi
5200:482
5161:doi
5139:425
5054:doi
5050:203
5015:doi
4856:of
4673:If
4628:is
4327:is
4056:in
3827:r,s
3764:of
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3627:or
3562:or
3342:of
3063:In
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2629:).
2609:In
2591:of
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1070:of
765:Hom
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209:or
181:,
149:to
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6058:.
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6018:.
6012:.
5992:.
5978:.
5974:.
5960:MR
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5931:MR
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5890:,
5869:MR
5867:,
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5814:MR
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5738:,
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5702:;
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5574:^
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2172:RP
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2055:SO
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