25:
810:
For a connected manifold, "open" is equivalent to "without boundary and non-compact", but for a disconnected manifold, open is stronger. For instance, the disjoint union of a circle and a line is non-compact since a line is non-compact, but this is not an open manifold since the circle (one of its
827:(along with some other technical conditions), thus by this definition a manifold does not include its boundary when it is embedded in a larger space. However, this definition doesnât cover some basic objects such as a
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866:" can refer to the universe being a closed manifold but more likely refers to the universe being a manifold of constant positive
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Every closed manifold is a
Euclidean neighborhood retract and thus has finitely generated homology groups.
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854:. A line is a closed subset of the plane, and a manifold, but not a closed manifold.
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is a compact two-dimensional manifold, but it is not closed because it has a boundary.
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843:(compact with respect to its underlying topology) can synonymously be used for
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Volume 1. 3rd edition with corrections. Publish or Perish, Houston TX 2005,
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795:{\displaystyle H^{k}(M;\mathbb {Z} _{2})\cong H_{n-k}(M;\mathbb {Z} _{2})}
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or not. Moreover, the torsion subgroup of the (n-1)-th homology group
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Most books generally define a manifold as a space that is, locally,
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is orientable or not. This follows from an application of the
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The notion of a closed manifold is unrelated to that of a
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is a closed connected n-manifold, the n-th homology group
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839:without reference to the boundary. But normally, a
49:. Unsourced material may be challenged and removed.
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708:-orientable. So there is always an isomorphism
847:if the usual definition for manifold is used.
183:are all closed two-dimensional manifolds. The
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151:is a manifold without boundary that has only
969:Cambridge University Press, Cambridge, 2002.
621:{\displaystyle D:H^{k}(M;R)\to H_{n-k}(M;R)}
201:is not closed because it is not compact. A
123:Classification of manifolds § Point-set
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981:
679:. In particular, every closed manifold is
675:is an isomorphism for all k. This is the
779:
736:
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109:Learn how and when to remove this message
1002:
190:is a closed n-dimensional manifold. The
121:For broader coverage of this topic, see
668:{\displaystyle D(\alpha )=\cap \alpha }
369:{\displaystyle H_{n-1}(M;\mathbb {Z} )}
197:is a closed 2n-dimensional manifold. A
1852:
976:
273:{\displaystyle H_{n}(M;\mathbb {Z} )}
814:
47:adding citations to reliable sources
18:
13:
16:Topological concept in mathematics
14:
1881:
857:
805:
831:, so authors sometimes define a
701:{\displaystyle \mathbb {Z} _{2}}
398:{\displaystyle \mathbb {Z} _{2}}
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34:needs additional citations for
1035:Differentiable/Smooth manifold
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539:{\displaystyle \in H_{n}(M;R)}
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1:
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427:universal coefficient theorem
208:
167:one-dimensional example is a
295:{\displaystyle \mathbb {Z} }
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1741:Classification of manifolds
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452:be a commutative ring. For
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10:
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302:or 0 depending on whether
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1817:over commutative algebras
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1533:Riemann curvature tensor
811:components) is compact.
192:complex projective space
934:See Hatcher 2002, p.250
925:See Hatcher 2002, p.238
916:See Hatcher 2002, p.236
907:See Hatcher 2002, p.536
898:See Hatcher 2002, p.231
492:with fundamental class
1325:Manifold with boundary
1040:Differential structure
833:manifold with boundary
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1860:Differential geometry
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405:depending on whether
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185:real projective space
1472:Covariant derivative
1023:Topological manifold
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147:. In comparison, an
43:improve this article
1506:Exterior derivative
1108:AtiyahâSinger index
1057:Riemannian manifold
966:Algebraic Topology.
1870:Geometric topology
1812:Secondary calculus
1766:Singularity theory
1721:Parallel transport
1489:De Rham cohomology
1128:Generalized Stokes
835:and abusively say
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1494:Differential form
1148:Whitney embedding
1082:Differential form
862:The notion of a "
815:Abuse of language
485:{\displaystyle M}
465:{\displaystyle R}
445:{\displaystyle R}
418:{\displaystyle M}
315:{\displaystyle M}
229:{\displaystyle M}
119:
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58:"Closed manifold"
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1839:Stratified space
1797:Fréchet manifold
1511:Interior product
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841:compact manifold
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677:Poincaré duality
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141:without boundary
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1782:Banach manifold
1775:Generalizations
1770:
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1521:Ricci curvature
1477:Cotangent space
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1188:Exponential map
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868:Ricci curvature
864:closed universe
860:
845:closed manifold
825:Euclidean space
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134:closed manifold
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1335:Parallelizable
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1215:Lie derivative
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1210:Integral curve
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1171:Diffeomorphism
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1016:Basic concepts
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943:Michael Spivak
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858:Use in physics
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1834:Supermanifold
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1555:Wedge product
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1499:Vector-valued
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1429:Tangent space
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1198:
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1193:in Lie theory
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1096:Main results
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1077:Tangent space
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962:
961:Allen Hatcher
959:
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955:0-914098-70-5
952:
948:
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904:
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880:Tame manifold
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149:open manifold
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60: â
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
1761:Moving frame
1756:Morse theory
1746:Gauge theory
1538:Tensor field
1467:Closed/Exact
1446:Vector field
1414:Distribution
1355:Hypercomplex
1350:Quaternionic
1242:
1087:Vector field
1045:Smooth atlas
965:
946:
930:
921:
912:
903:
894:
861:
849:
844:
840:
836:
821:homeomorphic
818:
809:
472:-orientable
431:
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187:
181:Klein bottle
162:
155:components.
152:
148:
133:
127:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
1706:Levi-Civita
1696:Generalized
1668:Connections
1618:Lie algebra
1550:Volume form
1451:Vector flow
1424:Pushforward
1419:Lie bracket
1318:Lie algebra
1283:G-structure
1072:Pushforward
1052:Submanifold
829:closed disk
628:defined by
203:closed disk
153:non-compact
130:mathematics
1854:Categories
1829:Stratifold
1787:Diffeology
1583:Associated
1384:Symplectic
1369:Riemannian
1298:Hyperbolic
1225:Submersion
1133:HopfâRinow
1067:Submersion
1062:Smooth map
886:References
852:closed set
546:, the map
324:orientable
209:Properties
179:, and the
99:March 2023
69:newspapers
1865:Manifolds
1711:Principal
1686:Ehresmann
1643:Subbundle
1633:Principal
1608:Fibration
1588:Cotangent
1460:Covectors
1313:Lie group
1293:Hermitian
1236:manifolds
1205:Immersion
1200:Foliation
1138:Noether's
1123:Frobenius
1118:De Rham's
1113:Darboux's
1004:Manifolds
761:−
750:≅
663:α
660:∩
642:α
596:−
585:→
509:∈
342:−
165:connected
163:The only
1807:Orbifold
1802:K-theory
1792:Diffiety
1516:Pullback
1330:Oriented
1308:Kenmotsu
1288:Hadamard
1234:Types of
1183:Geodesic
1008:Glossary
874:See also
837:manifold
376:is 0 or
159:Examples
143:that is
138:manifold
1751:History
1734:Related
1648:Tangent
1626:)
1606:)
1573:Adjoint
1565:Bundles
1543:density
1441:Torsion
1407:Vectors
1399:Tensors
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1363:,
1361:Pseudoâ
1340:Poisson
1273:Finsler
1268:Fibered
1263:Contact
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1253:Complex
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1220:Section
171:. The
145:compact
83:scholar
1716:Vector
1701:Koszul
1681:Cartan
1676:Affine
1658:Vector
1653:Tensor
1638:Spinor
1628:Normal
1624:Stable
1578:Affine
1482:bundle
1434:bundle
1380:Almost
1303:KĂ€hler
1259:Almost
1249:Almost
1243:Closed
1143:Sard's
1099:(list)
953:
173:sphere
169:circle
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1824:Sheaf
1598:Fiber
1374:Rizza
1345:Prime
1176:Local
1166:Curve
1028:Atlas
177:torus
136:is a
90:JSTOR
76:books
1691:Form
1593:Dual
1526:flow
1389:Tame
1365:Subâ
1278:Flat
1158:Maps
951:ISBN
432:Let
199:line
132:, a
62:news
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823:to
322:is
280:is
216:If
128:In
45:by
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