279:
The prime decomposition holds also for non-orientable 3-manifolds, but the uniqueness statement must be modified slightly. Every compact, non-orientable 3-manifold is a connected sum of irreducible 3-manifolds and non-orientable
220:, which means that any embedded 2-sphere bounds a ball. So the theorem can be restated to say that there is a unique connected sum decomposition into irreducible 3-manifolds and fiber bundles of
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if it is not homeomorphic to any connected sum of manifolds, except for the trivial connected sum of the manifold with a sphere of the same dimension,
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410:. Existence was proven by Kneser, but the exact formulation and proof of the uniqueness was done more than 30 years later by
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This sum is unique as long as we specify that each summand is either irreducible or a non-orientable
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526:"Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten"
428:. Annals of Mathematics Studies. Vol. 86. Princeton, NJ:
562:(1962). "A unique decomposition theorem for 3-manifolds".
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Jahresbericht der
Deutschen Mathematiker-Vereinigung
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22:prime decomposition theorem for 3-manifolds
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116:is a prime 3-manifold then either it is
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476:Lectures on three-manifold topology
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14:
1506:Theorems in differential geometry
149:{\displaystyle S^{2}\times S^{1}}
565:American Journal of Mathematics
671:Differentiable/Smooth manifold
1:
480:American Mathematical Society
89:{\textstyle M\cong M\#S^{n}}
1377:Classification of manifolds
1522:
430:Princeton University Press
1453:over commutative algebras
544:10.1515/9783110894516.147
406:techniques originated by
1169:Riemann curvature tensor
46:) finite collection of
961:Manifold with boundary
676:Differential structure
402:The proof is based on
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393:{\displaystyle S^{1}.}
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333:{\displaystyle S^{1}.}
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270:{\displaystyle S^{1}.}
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209:{\displaystyle S^{1},}
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156:or the non-orientable
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110:
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424:Hempel, John (1976).
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362:
360:{\displaystyle S^{2}}
335:
302:
300:{\displaystyle S^{2}}
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242:
240:{\displaystyle S^{2}}
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178:
176:{\displaystyle S^{2}}
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1108:Covariant derivative
659:Topological manifold
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61:
1142:Exterior derivative
744:Atiyah–Singer index
693:Riemannian manifold
1448:Secondary calculus
1402:Singularity theory
1357:Parallel transport
1125:De Rham cohomology
764:Generalized Stokes
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24:states that every
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1130:Differential form
784:Whitney embedding
718:Differential form
109:{\displaystyle P}
48:prime 3-manifolds
1513:
1475:Stratified space
1433:Fréchet manifold
1147:Interior product
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522:Kneser, Hellmuth
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1418:Banach manifold
1411:Generalizations
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1157:Ricci curvature
1113:Cotangent space
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824:Exponential map
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971:Parallelizable
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851:Lie derivative
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652:Basic concepts
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53:A manifold is
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1470:Supermanifold
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1135:Vector-valued
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1065:Tangent space
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829:in Lie theory
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732:Main results
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713:Tangent space
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497:0-8218-1693-4
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485:
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472:Jaco, William
469:
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447:0-8218-3695-1
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45:
44:homeomorphism
42:
39:of a unique (
38:
37:connected sum
34:
31:
27:
23:
19:
1397:Moving frame
1392:Morse theory
1382:Gauge theory
1174:Tensor field
1103:Closed/Exact
1082:Vector field
1050:Distribution
991:Hypercomplex
986:Quaternionic
723:Vector field
681:Smooth atlas
569:
563:
535:
529:
475:
425:
401:
278:
54:
52:
21:
15:
1496:3-manifolds
1342:Levi-Civita
1332:Generalized
1304:Connections
1254:Lie algebra
1186:Volume form
1087:Vector flow
1060:Pushforward
1055:Lie bracket
954:Lie algebra
919:G-structure
708:Pushforward
688:Submanifold
538:: 248–259.
426:3-Manifolds
412:John Milnor
218:irreducible
18:mathematics
1490:Categories
1465:Stratifold
1423:Diffeology
1219:Associated
1020:Symplectic
1005:Riemannian
934:Hyperbolic
861:Submersion
769:Hopf–Rinow
703:Submersion
698:Smooth map
602:0108.36501
572:(1): 1–7.
560:Milnor, J.
552:55.0311.03
514:0433.57001
464:0345.57001
418:References
33:3-manifold
30:orientable
1501:Manifolds
1347:Principal
1322:Ehresmann
1279:Subbundle
1269:Principal
1244:Fibration
1224:Cotangent
1096:Covectors
949:Lie group
929:Hermitian
872:manifolds
841:Immersion
836:Foliation
774:Noether's
759:Frobenius
754:De Rham's
749:Darboux's
640:Manifolds
594:122595895
216:or it is
134:×
74:#
68:≅
1443:Orbifold
1438:K-theory
1428:Diffiety
1152:Pullback
966:Oriented
944:Kenmotsu
924:Hadamard
870:Types of
819:Geodesic
644:Glossary
524:(1929).
474:(1980).
1387:History
1370:Related
1284:Tangent
1262:)
1242:)
1209:Adjoint
1201:Bundles
1179:density
1077:Torsion
1043:Vectors
1035:Tensors
1018:)
1003:)
999:,
997:Pseudo−
976:Poisson
909:Finsler
904:Fibered
899:Contact
897:)
889:Complex
887:)
856:Section
586:0142125
506:0565450
456:0415619
308:bundles
35:is the
26:compact
1352:Vector
1337:Koszul
1317:Cartan
1312:Affine
1294:Vector
1289:Tensor
1274:Spinor
1264:Normal
1260:Stable
1214:Affine
1118:bundle
1070:bundle
1016:Almost
939:Kähler
895:Almost
885:Almost
879:Closed
779:Sard's
735:(list)
600:
592:
584:
550:
512:
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494:
462:
454:
444:
368:bundle
184:bundle
20:, the
1460:Sheaf
1234:Fiber
1010:Rizza
981:Prime
812:Local
802:Curve
664:Atlas
590:S2CID
370:over
310:over
247:over
186:over
96:. If
55:prime
41:up to
1327:Form
1229:Dual
1162:flow
1025:Tame
1001:Sub−
914:Flat
794:Maps
492:ISBN
442:ISBN
1249:Jet
598:Zbl
574:doi
548:JFM
540:doi
510:Zbl
484:doi
460:Zbl
434:doi
16:In
1492::
1240:Co
596:.
588:.
582:MR
580:.
570:84
568:.
546:.
536:38
534:.
528:.
508:.
502:MR
500:.
490:.
482:.
458:.
452:MR
450:.
440:.
432:.
414:.
50:.
28:,
1258:(
1238:(
1014:(
995:(
893:(
883:(
646:)
642:(
632:e
625:t
618:v
604:.
576::
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542::
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486::
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436::
388:.
383:1
379:S
353:2
349:S
328:.
323:1
319:S
293:2
289:S
265:.
260:1
256:S
233:2
229:S
204:,
199:1
195:S
169:2
165:S
142:1
138:S
129:2
125:S
104:P
82:n
78:S
71:M
65:M
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