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2017:
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is a locally compact
Hausdorff space, hence it is (completely) regular. Assume such a space X is σ-compact. Then it is Lindelöf, and because Lindelöf + regular implies paracompact, X is metrizable. But in a metrizable space, second-countability coincides with being Lindelöf, so X is second-countable.
557:
1-dimensional topological manifold that is not metrizable nor paracompact. Since metrizability is such a desirable property for a topological space, it is common to add paracompactness to the definition of a manifold. In any case, non-paracompact manifolds are generally regarded as
1470:
1108:
72:). Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure. However, not every topological manifold can be endowed with a particular additional structure. For example, the
922:
1875:Žubr A.V. (1988) Classification of simply-connected topological 6-manifolds. In: Viro O.Y., Vershik A.M. (eds) Topology and Geometry — Rohlin Seminar. Lecture Notes in Mathematics, vol 1346. Springer, Berlin, Heidelberg
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of the disjoint union of the resulting manifolds with boundary, with the quotient taken with regards to a homeomorphism between the boundary spheres of the removed balls. This results in another
1797:
1142:. That is, coordinate charts agree on overlaps up to homeomorphism. Different types of manifolds can be defined by placing restrictions on types of transition maps allowed. For example, for
530:
points, an open neighborhood of either of which includes all nonzero numbers in some open interval centered at zero. This space is not
Hausdorff because the two origins cannot be separated.
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816:
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592:
Every second-countable manifold is paracompact, but not vice versa. However, the converse is nearly true: a paracompact manifold is second-countable if and only if it has a
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Adding the
Hausdorff condition can make several properties become equivalent for a manifold. As an example, we can show that for a Hausdorff manifold, the notions of
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Barden, D. "Simply
Connected Five-Manifolds." Annals of Mathematics, vol. 82, no. 3, 1965, pp. 365–385. JSTOR, www.jstor.org/stable/1970702.
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that
Euclidean neighborhoods are always open sets. One can always find Euclidean neighborhoods that are homeomorphic to "nice" open sets in
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in 2003. More specifically, Perelman's results provide an algorithm for deciding if two three-manifolds are homeomorphic to each other.
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1999:
1974:
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are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold (e.g.
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600:. In particular, a connected manifold is paracompact if and only if it is second-countable. Every second-countable manifold is
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597:
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136:. It is common to place additional requirements on topological manifolds. In particular, many authors define them to be
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The
Hausdorff property is not a local one; so even though Euclidean space is Hausdorff, a locally Euclidean space need
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60:. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All
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By definition, every point of a locally
Euclidean space has a neighborhood homeomorphic to an open subset of
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and paracompact. Moreover, if a manifold is separable and paracompact then it is also second-countable.
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in some finite-dimensional
Euclidean space. For any manifold the properties of being second-countable,
566:. Paracompact manifolds have all the topological properties of metric spaces. In particular, they are
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since manifolds are locally-connected. Being locally path connected, a manifold is path-connected
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1465:{\displaystyle \mathbb {R} _{+}^{n}=\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}:x_{n}\geq 0\}.}
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1103:{\displaystyle \psi \phi ^{-1}:\phi \left(U\cap V\right)\rightarrow \psi \left(U\cap V\right)}
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Manifolds inherit many of the local properties of
Euclidean space. In particular, they are
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Proceedings of the
International Congress of Mathematicians: Hyderabad, August 19-27, 2010
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will mean a topological manifold such that every point has a neighborhood homeomorphic to
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is locally Euclidean if and only if either of the following equivalent conditions holds:
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Every topological manifold is a topological manifold with boundary, but not vice versa.
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1324:. There is, however, a classification of simply connected manifolds of dimension ≥ 5.
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it is connected. It follows that the path-components are the same as the components.
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in which every point has a neighborhood homeomorphic to an open subset of Euclidean
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is a topological manifold which cannot be endowed with a differentiable structure.
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1572:-manifolds is defined by removing an open ball from each manifold and taking the
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Conversely, if X is a Hausdorff second-countable manifold, it must be σ-compact.
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917:{\displaystyle \phi :U\rightarrow \phi \left(U\right)\subset \mathbb {R} ^{n}}
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1963:
Foundational Essays on Topological Manifolds. Smoothings, and Triangulations
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greater than three is known to be impossible; it is at least as hard as the
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Every nonempty, paracompact, connected 1-manifold is homeomorphic either to
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is frequently used to refer to the domain or range of such a map). A space
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is a topological manifold which cannot be given a differentiable structure.
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by Euclidean neighborhoods. A set of Euclidean neighborhoods that cover
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There are several methods of creating manifolds from other manifolds.
526:. This space is created by replacing the origin of the real line with
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457:. Being locally compact Hausdorff spaces, manifolds are necessarily
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are a class of differentiable manifolds equipped with a compatible
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Hausdorff locally compact and second countable is sigma-compact
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An example of a non-Hausdorff locally Euclidean space is the
512:. It is true, however, that every locally Euclidean space is
1928:
Gauld, D. B. (1974). "Topological Properties of Manifolds".
1173:. A discrete space is second-countable if and only if it is
562:. An example of a non-paracompact manifold is given by the
1799:
A Guide to the Classification Theorem for Compact Surfaces
1550:-manifold (the pieces must all have the same dimension).
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A Euclidean neighborhood homeomorphic to an open ball in
655:, meaning that any topological space homeomorphic to an
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The property of being locally Euclidean is preserved by
1855:. Springer Science & Business Media. pp. 90–.
1775:. Springer Science & Business Media. pp. 64–.
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A slightly more general concept is sometimes useful. A
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Such a map is a homeomorphism between open subsets of
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A manifold need not be connected, but every manifold
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are a class of topological manifolds equipped with a
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Locally compact Hausdorff implies completely regular
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1936:(6). Mathematical Association of America: 633–636.
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1229:Every nonempty, compact, connected 2-manifold (or
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368:are a class of differentiable manifolds that are
1737:. European Mathematical Society. pp. 249–.
468:and second-countability are the same. Indeed, a
859:for the topology of a locally Euclidean space.
958:. (The terminology comes from an analogy with
940:is locally Euclidean if and only if it can be
611:manifold is second-countable and paracompact.
430:. In particular, being locally Euclidean is a
333:Manifolds related to projective space include
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1263:A classification of 3-manifolds results from
8:
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1796:Jean Gallier; Dianna Xu (5 February 2013).
1686:. American Mathematical Soc. pp. 25–.
573:Manifolds are also commonly required to be
484:of connected manifolds. These are just the
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2050:
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1903:. American Mathematical Soc. pp. 7–.
68:are topological manifolds equipped with a
1969:. Princeton: Princeton University Press.
1802:. Springer Science & Business Media.
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1659:. Springer Science & Business Media.
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1852:Differentiable Manifolds: A First Course
1828:. European Mathematical Society. 2010.
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1146:the transition maps are required to be
1897:Jeffrey Lee; Jeffrey Marc Lee (2009).
1334:Manifold § Manifold with boundary
749:has a neighborhood homeomorphic to an
1987:Introduction to Topological Manifolds
1772:Introduction to Topological Manifolds
1656:Introduction to Topological Manifolds
151:will mean a topological manifold. An
7:
1265:Thurston's geometrization conjecture
1900:Manifolds and Differential Geometry
1629:. World Scientific. pp. 477–.
1213:Classification theorem for surfaces
789:has a neighborhood homeomorphic to
534:Compactness and countability axioms
147:In the remainder of this article a
1960:; Siebenmann, Laurence C. (1977).
1340:topological manifold with boundary
426:is locally Euclidean of dimension
406:is locally Euclidean of dimension
25:
1930:The American Mathematical Monthly
1849:Lawrence Conlon (17 April 2013).
1683:A Course in Differential Geometry
568:perfectly normal Hausdorff spaces
2920:Properties of topological spaces
2015:
1990:. Graduate Texts in Mathematics
1135:{\displaystyle \mathbb {R} ^{n}}
844:{\displaystyle \mathbb {R} ^{n}}
811:{\displaystyle \mathbb {R} ^{n}}
775:{\displaystyle \mathbb {R} ^{n}}
730:{\displaystyle \mathbb {R} ^{n}}
700:. Such neighborhoods are called
693:{\displaystyle \mathbb {R} ^{n}}
1623:Rajendra Bhatia (6 June 2011).
862:For any Euclidean neighborhood
639:. The dimension of a non-empty
422:is a local homeomorphism, then
2090:Differentiable/Smooth manifold
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1:
1825:Geometrisation of 3-manifolds
1769:John Lee (25 December 2010).
319:Quaternionic projective space
1653:John M. Lee (6 April 2006).
1159:Discrete spaces (0-Manifold)
222:are compact 2-manifolds (or
200:is a 0-dimensional manifold.
2796:Classification of manifolds
1311:algorithmically undecidable
1293:The full classification of
1154:Classification of manifolds
372:of odd-dimensional spheres.
95:if there is a non-negative
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1526:)-manifold when given the
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2872:over commutative algebras
1542:of a countable family of
1233:) is homeomorphic to the
1013:with overlapping domains
970:of flat maps or charts).
855:. Euclidean balls form a
102:such that every point in
27:Type of topological space
2588:Riemann curvature tensor
1731:Tammo tom Dieck (2008).
1245:, or a connected sum of
1144:differentiable manifolds
356:Differentiable manifolds
306:Complex projective space
66:differentiable manifolds
1328:Manifolds with boundary
1309:, which is known to be
1169:A 0-manifold is just a
966:can be described by an
702:Euclidean neighborhoods
627:-manifold cannot be an
289:are compact manifolds.
132:is a locally Euclidean
46:that locally resembles
2380:Manifold with boundary
2095:Differential structure
2022:Mathematical manifolds
1994:. New York: Springer.
1680:Thierry Aubin (2001).
1594:Any open subset of an
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328:-dimensional manifold.
315:-dimensional manifold.
302:-dimensional manifold.
260:circles) is a compact
70:differential structure
1984:Lee, John M. (2000).
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1316:In fact, there is no
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1203:Surfaces (2-Manifold)
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1006:{\displaystyle \psi }
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986:{\displaystyle \phi }
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659:-manifold is also an
542:if and only if it is
524:line with two origins
293:Real projective space
184:real coordinate space
2527:Covariant derivative
2078:Topological manifold
2024:at Wikimedia Commons
1706:Topospaces subwiki,
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1253:Volumes (3-Manifold)
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962:whereby a spherical
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706:invariance of domain
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653:topological property
621:invariance of domain
598:connected components
589:are all equivalent.
486:connected components
451:locally contractible
432:topological property
400:local homeomorphisms
269:Projective manifolds
130:topological manifold
40:topological manifold
2561:Exterior derivative
2163:Atiyah–Singer index
2112:Riemannian manifold
1602:-manifold with the
1380:
1181:Curves (1-Manifold)
1023:transition function
932:(although the word
504:The Hausdorff axiom
234:-dimensional sphere
2867:Secondary calculus
2821:Singularity theory
2776:Parallel transport
2544:De Rham cohomology
2183:Generalized Stokes
1734:Algebraic Topology
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866:, a homeomorphism
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737:. Indeed, a space
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470:Hausdorff manifold
455:locally metrizable
251:-dimensional torus
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2549:Differential form
2203:Whitney embedding
2137:Differential form
2020:Media related to
1910:978-0-8218-4815-9
1862:978-1-4757-2284-0
1835:978-3-03719-082-1
1809:978-3-642-34364-3
1782:978-1-4419-7940-7
1744:978-3-03719-048-7
1693:978-0-8218-7214-7
1666:978-0-387-22727-6
1636:978-981-4324-35-9
1604:subspace topology
1509:Cartesian product
1487:Product manifolds
1247:projective planes
973:Given two charts
667:Coordinate charts
443:locally connected
343:Stiefel manifolds
275:Projective spaces
169:List of manifolds
93:locally Euclidean
86:topological space
80:Formal definition
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16:(Redirected from
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2071:Basic concepts
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2010:External links
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616:
615:Dimensionality
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538:A manifold is
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498:if and only if
482:disjoint union
402:. That is, if
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2816:Moving frame
2811:Morse theory
2801:Gauge theory
2593:Tensor field
2522:Closed/Exact
2501:Vector field
2469:Distribution
2410:Hypercomplex
2405:Quaternionic
2142:Vector field
2100:Smooth atlas
2077:
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1337:
1315:
1307:group theory
1303:word problem
1298:
1294:
1292:
1276:
1267:, proven by
1262:
1228:
1192:
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1168:
1112:
1022:
1018:
1014:
972:
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851:is called a
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492:, which are
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220:Klein bottle
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129:
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122:
117:
112:homeomorphic
108:neighborhood
103:
99:
92:
88:
83:
50:
39:
29:
2761:Levi-Civita
2751:Generalized
2723:Connections
2673:Lie algebra
2605:Volume form
2506:Vector flow
2479:Pushforward
2474:Lie bracket
2373:Lie algebra
2338:G-structure
2127:Pushforward
2107:Submanifold
1590:Submanifold
1584:Submanifold
1580:-manifold.
960:cartography
663:-manifold.
647:. Being an
544:paracompact
387:E8 manifold
366:Lens spaces
287:quaternions
211:1-manifold.
138:paracompact
74:E8 manifold
55:dimensional
36:mathematics
2909:Categories
2884:Stratifold
2842:Diffeology
2638:Associated
2439:Symplectic
2424:Riemannian
2353:Hyperbolic
2280:Submersion
2188:Hopf–Rinow
2122:Submersion
2117:Smooth map
1922:References
1348:half-space
1289:5-manifold
1285:4-manifold
1259:3-manifold
1209:2-manifold
1187:1-manifold
596:number of
540:metrizable
394:Properties
382:structure.
376:Lie groups
264:-manifold.
243:-manifold.
193:-manifold.
177:-Manifolds
153:n-manifold
91:is called
2915:Manifolds
2766:Principal
2741:Ehresmann
2698:Subbundle
2688:Principal
2663:Fibration
2643:Cotangent
2515:Covectors
2368:Lie group
2348:Hermitian
2291:manifolds
2260:Immersion
2255:Foliation
2193:Noether's
2178:Frobenius
2173:De Rham's
2168:Darboux's
2059:Manifolds
1610:Footnotes
1451:≥
1423:∈
1404:…
1318:algorithm
1279:-manifold
1175:countable
1090:∩
1079:ψ
1076:→
1065:∩
1054:ϕ
1043:−
1039:ϕ
1035:ψ
1001:ψ
981:ϕ
900:⊂
886:ϕ
883:→
874:ϕ
751:open ball
602:separable
594:countable
587:σ-compact
564:long line
555:Hausdorff
548:long line
494:open sets
370:quotients
283:complexes
277:over the
110:which is
62:manifolds
2862:Orbifold
2857:K-theory
2847:Diffiety
2571:Pullback
2385:Oriented
2363:Kenmotsu
2343:Hadamard
2289:Types of
2238:Geodesic
2063:Glossary
1574:quotient
1275:General
583:Lindelöf
414: :
224:surfaces
163:Examples
149:manifold
32:topology
2806:History
2789:Related
2703:Tangent
2681:)
2661:)
2628:Adjoint
2620:Bundles
2598:density
2496:Torsion
2462:Vectors
2454:Tensors
2437:)
2422:)
2418:,
2416:Pseudo−
2395:Poisson
2328:Finsler
2323:Fibered
2318:Contact
2316:)
2308:Complex
2306:)
2275:Section
1950:2319220
1568:of two
1231:surface
1195:or the
942:covered
818:itself.
609:compact
209:compact
97:integer
2771:Vector
2756:Koszul
2736:Cartan
2731:Affine
2713:Vector
2708:Tensor
2693:Spinor
2683:Normal
2679:Stable
2633:Affine
2537:bundle
2489:bundle
2435:Almost
2358:Kähler
2314:Almost
2304:Almost
2298:Closed
2198:Sard's
2154:(list)
1998:
1973:
1948:
1907:
1859:
1832:
1806:
1779:
1741:
1690:
1663:
1633:
1518:is a (
1514:×
1503:is an
1495:is an
1235:sphere
1223:sphere
1197:circle
1148:smooth
607:Every
585:, and
579:embeds
552:normal
546:. The
510:not be
453:, and
341:, and
324:is a 4
311:is a 2
218:and a
205:circle
189:is an
120:-space
106:has a
2879:Sheaf
2653:Fiber
2429:Rizza
2400:Prime
2231:Local
2221:Curve
2083:Atlas
1967:(PDF)
1946:JSTOR
1342:is a
968:atlas
964:globe
951:atlas
934:chart
857:basis
480:is a
380:group
298:is a
285:, or
279:reals
216:torus
207:is a
116:real
42:is a
2746:Form
2648:Dual
2581:flow
2444:Tame
2420:Sub−
2333:Flat
2213:Maps
1996:ISBN
1971:ISBN
1905:ISBN
1857:ISBN
1830:ISBN
1804:ISBN
1777:ISBN
1739:ISBN
1688:ISBN
1661:ISBN
1631:ISBN
1564:The
1538:The
1287:and
1243:tori
1237:, a
1221:The
1211:and
1017:and
993:and
410:and
385:The
246:The
229:The
196:Any
182:The
48:real
38:, a
2668:Jet
1992:202
1938:doi
1491:If
1354:):
1305:in
1241:of
954:on
928:on
753:in
619:By
528:two
488:of
140:or
114:to
30:In
2911::
2659:Co
1944:.
1934:81
1932:.
1889:^
1753:^
1645:^
1606:.
1530:.
1313:.
1249:.
1199:.
1177:.
1150:.
635:≠
570:.
519:.
461:.
449:,
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1600:n
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768:n
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326:n
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232:n
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157:R
123:R
118:n
104:X
100:n
89:X
53:-
51:n
20:)
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