1693:
1476:
270:
1714:
1682:
149:
1751:
1724:
1704:
29:
183:– because there are not enough dimensions, a Whitney disk introduces new kinks, which can be resolved by another Whitney disk, leading to a sequence ("tower") of disks. The limit of this tower yields a topological but not differentiable map, hence surgery works topologically but not differentiably in dimension 4.
1044:
are complex curves) – by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can be studied from the point of view of complex geometry in two variables (complex surfaces), though not every 4-manifold admits a complex structure.
168:
of a disk – the disk has 2 dimensions, and the homotopy adds 1 more – and thus in codimension greater than 2, this can be done without intersecting itself; hence embeddings in codimension greater than 2 can be understood by surgery. In surgery theory, the key step is in the middle dimension, and thus
144:
works in dimension 5 and above (in fact, in many cases, it works topologically in dimension 4, though this is very involved to prove), and thus the behavior of manifolds in dimension 5 and above may be studied using the surgery theory program. In dimension 4 and below (topologically, in dimension 3
238:
orientable manifold has exactly two different possible orientations. In this setting, various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds often employ methods of
412:
129:. Thus the topological classification of 4-manifolds is in principle tractable, and the key questions are: does a topological manifold admit a differentiable structure, and if so, how many? Notably, the smooth case of dimension 4 is the last open case of the
145:
and below), surgery theory does not work. Indeed, one approach to discussing low-dimensional manifolds is to ask "what would surgery theory predict to be true, were it to work?" – and then understand low-dimensional phenomena as deviations from this.
121:
Dimension 4 is special, in that in some respects (topologically), dimension 4 is high-dimensional, while in other respects (differentiably), dimension 4 is low-dimensional; this overlap yields phenomena exceptional to dimension 4, such as
164:, the key technical trick which underlies surgery theory, requires 2+1 dimensions. Roughly, the Whitney trick allows one to "unknot" knotted spheres – more precisely, remove self-intersections of immersions; it does this via a
1066:. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In mathematical language, a knot is an
1028:
in 2 dimensions – every surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive curvature/spherical, zero curvature/flat, negative curvature/hyperbolic – and the
1202:). Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along the cut or boundary. This is closely related to, but not identical with,
757:
212:
of a manifold is a very important invariant, and determines much of the structure; in dimensions 1, 2 and 3, the possible fundamental groups are restricted, while in dimension 4 and above every
298:
169:
when the middle dimension has codimension more than 2 (loosely, 2½ is enough, hence total dimension 5 is enough), the
Whitney trick works. The key consequence of this is Smale's
216:
is the fundamental group of a manifold (note that it is sufficient to show this for 4- and 5-dimensional manifolds, and then to take products with spheres to get higher ones).
907:
815:
786:
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is to a topological space—in many regards the purpose of a handle decomposition is to have a language analogous to CW-complexes, but adapted to the world of
251:. An important generalization of the notion of orientability of a space is that of orientability of a family of spaces parameterized by some other space (a
1754:
1094:); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.
1170:
which measure the deviation of a local product structure from a global product structure. They are one of the unifying geometric concepts in
255:) for which an orientation must be selected in each of the spaces which varies continuously with respect to changes in the parameter values.
1097:
To gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other
1033:(now theorem) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of 8 possible geometries.
712:
1388:
1162:. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses
130:
1742:
1737:
1360:
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192:
407:{\displaystyle \emptyset =M_{-1}\subset M_{0}\subset M_{1}\subset M_{2}\subset \dots \subset M_{m-1}\subset M_{m}=M}
1732:
1283:
1634:
110:
High-dimensional topology refers to manifolds of dimension 5 and above, or in relative terms, embeddings in
1780:
99:
theory. The use of the term geometric topology to describe these seems to have originated rather recently.
918:
1775:
20:
1082:(since we're using topology, a circle isn't bound to the classical geometric concept, but to all of its
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1086:). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of
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883:
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1206:. It is a major tool in the study and classification of manifolds of dimension greater than 3.
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is concerned with questions in dimensions up to 4, or embeddings in codimension up to 2.
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1703:
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1244:) led to the emergence of surgery theory as a major tool in high-dimensional topology.
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way (that is, the embedding extends to that of a thickened sphere), then the pair (
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563:
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of topological manifolds, locally flat submanifolds play a role similar to that of
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1334:
1315:
1233:
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1041:
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240:
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93:
37:
33:
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176:, which works in dimension 5 and above, and forms the basis for surgery theory.
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A modification of the
Whitney trick can work in 4 dimensions, and is called
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111:
45:
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1209:
More technically, the idea is to start with a well-understood manifold
1155:
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502:
77:
1217:′ having some desired property, in such a way that the effects on the
1588:
1067:
551:
123:
61:
1024:
Low-dimensional topology is strongly geometric, as reflected in the
160:
The precise reason for the difference at dimension 5 is because the
156:
requires 2+1 dimensions, hence surgery theory requires 5 dimensions.
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1573:
1541:
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1397:
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107:
Manifolds differ radically in behavior in high and low dimension.
103:
Differences between low-dimensional and high-dimensional topology
1071:
932:
517:-cell. Handle decompositions of manifolds arise naturally via
1021:
each have their own theory, where there are some connections.
247:
more structure is present, allowing a formulation in terms of
16:
Branch of mathematics studying (smooth) functions of manifolds
1225:, or other interesting invariants of the manifold are known.
521:. The modification of handle structures is closely linked to
76:
may be said to have originated in the 1935 classification of
1366:
40:; these surfaces can be used as tools in geometric topology
1166:
or not. In other words, characteristic classes are global
230:
A manifold is orientable if it has a consistent choice of
1310:(1960), A proof of the generalized Schoenflies theorem.
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752:{\displaystyle (\mathbb {R} ^{n},\mathbb {R} ^{d})}
501:. A handle decomposition is to a manifold what a
1194:from another in a 'controlled' way, introduced by
1190:is a collection of techniques used to produce one
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1213:and perform surgery on it to produce a manifold
1101:and objects other than circles can be used; see
84:, which required distinguishing spaces that are
1241:
978:for their independent proofs of this theorem.
1382:
72:Geometric topology as an area distinct from
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1750:
1723:
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1329:Mazur, Barry, On embeddings of spheres.,
1036:2-dimensional topology can be studied as
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797:
768:
736:
721:
817:. That is, there exists a homeomorphism
268:
258:
147:
27:
974:-sphere. Brown and Mazur received the
273:A 3-ball with three 1-handles attached.
1768:
1199:
912:
1370:
513:-handle is the smooth analogue of an
187:Important tools in geometric topology
197:
124:exotic differentiable structures on
1122:High-dimensional geometric topology
13:
931: − 1)-dimensional
302:
14:
1792:
1314:, vol. 66, pp. 74–76.
1259:List of geometric topology topics
528:
193:List of geometric topology topics
19:For the mathematical object, see
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1722:
1712:
1702:
1691:
1681:
1680:
1474:
1143:is a way of associating to each
902:{\displaystyle \mathbb {R} ^{d}}
810:{\displaystyle \mathbb {R} ^{n}}
781:{\displaystyle \mathbb {R} ^{d}}
219:
1107:. Higher-dimensional knots are
958:) is homeomorphic to the pair (
759:, with a standard inclusion of
131:generalized Poincaré conjecture
1353:Handbook of Geometric Topology
1323:
1301:
1276:
1126:In high-dimensional topology,
1118:-dimensional Euclidean space.
1048:
982:Branches of geometric topology
827:
746:
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692:
674:
64:of one manifold into another.
1:
1284:"What is geometric topology?"
1269:
1396:
7:
1288:math.meta.stackexchange.com
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1130:are a basic invariant, and
698:{\displaystyle (U,U\cap N)}
634:if there is a neighborhood
140:The distinction is because
60:between them, particularly
21:Geometric topology (object)
10:
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1643:Banach fixed-point theorem
1254:Category:Maps of manifolds
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843:{\displaystyle U\to R^{n}}
653:{\displaystyle U\subset M}
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1031:geometrization conjecture
919:Jordan-Schönflies theorem
162:Whitney embedding theorem
92:. This was the origin of
1099:three-dimensional spaces
998:Low-dimensional topology
993:Low-dimensional topology
987:Low-dimensional topology
245:differentiable manifolds
214:finitely presented group
116:Low-dimensional topology
873:{\displaystyle U\cap N}
615:{\displaystyle x\in N,}
470:{\displaystyle M_{i-1}}
208:In all dimensions, the
1698:Mathematics portal
1598:Metrics and properties
1584:Second-countable space
1331:Bull. Amer. Math. Soc.
1312:Bull. Amer. Math. Soc.
1264:Plumbing (mathematics)
1228:The classification of
1128:characteristic classes
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437:{\displaystyle M_{i}}
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259:Handle decompositions
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1653:Invariance of domain
1605:Euler characteristic
1579:Bundle (mathematics)
1140:characteristic class
1112:-dimensional spheres
942:-dimensional sphere
927:states that, if an (
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577:is embedded into an
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477:by the attaching of
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279:handle decomposition
265:Handle decomposition
82:Reidemeister torsion
36:bounded by a set of
1781:Geometry processing
1663:Tychonoff's theorem
1658:Poincaré conjecture
1412:General (point-set)
925:Schoenflies theorem
913:Schönflies theorems
562:in the category of
542:is a property of a
86:homotopy equivalent
1776:Geometric topology
1648:De Rham cohomology
1569:Polyhedral complex
1559:Simplicial complex
1180:algebraic geometry
1172:algebraic topology
1104:knot (mathematics)
1064:mathematical knots
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249:differential forms
174:-cobordism theorem
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74:algebraic topology
50:geometric topology
42:
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1762:
1552:fundamental group
1355:, North-Holland.
1149:topological space
1134:is a key theory.
1074:in 3-dimensional
1040:in one variable (
788:as a subspace of
490:{\displaystyle i}
444:is obtained from
210:fundamental group
204:Fundamental group
198:Fundamental group
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1347:R. B. Sher and
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1132:surgery theory
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1084:homeomorphisms
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664:such that the
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535:Local flatness
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529:Local flatness
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181:Casson handles
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1755:Publications
1620:Chern number
1610:Betti number
1493: /
1484:Key concepts
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1432:Differential
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1291:. Retrieved
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948:locally flat
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709:to the pair
707:homeomorphic
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519:Morse theory
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135:Gluck twists
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90:homeomorphic
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1718:Wikiversity
1635:Key results
1060:Knot theory
1055:Knot theory
1049:Knot theory
1016:4-manifolds
1011:3-manifolds
544:submanifold
523:Cerf theory
509:. Thus an
417:where each
292:is a union
232:orientation
112:codimension
78:lens spaces
46:mathematics
1770:Categories
1564:CW complex
1505:Continuity
1495:Closed set
1454:cohomology
1270:References
1168:invariants
1156:cohomology
1000:includes:
569:Suppose a
550:of larger
62:embeddings
1743:geometric
1738:algebraic
1589:Cobordism
1525:Hausdorff
1520:connected
1437:Geometric
1427:Continuum
1417:Algebraic
1158:class of
1068:embedding
966:), where
865:∩
828:→
687:∩
645:⊂
604:∈
554:. In the
552:dimension
460:−
386:⊂
378:−
367:⊂
364:⋯
361:⊂
348:⊂
335:⊂
322:⊂
314:−
303:∅
236:connected
54:manifolds
1708:Wikibook
1686:Category
1574:Manifold
1542:Homotopy
1500:Interior
1491:Open set
1449:Homology
1398:Topology
1351:(2002),
1248:See also
1234:Kervaire
1219:homology
1192:manifold
1164:sections
1005:Surfaces
556:category
287:manifold
234:, and a
166:homotopy
97:homotopy
88:but not
1733:general
1535:uniform
1515:compact
1466:Digital
1338:0117693
1319:0117695
1293:May 30,
1240: (
1198: (
954:,
622:we say
593:). If
585:(where
499:handles
68:History
1728:Topics
1530:metric
1405:Fields
1359:
1238:Milnor
1196:Milnor
1072:circle
933:sphere
281:of an
133:; see
95:simple
1510:Space
1147:on a
1070:of a
946:in a
852:image
589:<
546:in a
1357:ISBN
1295:2018
1242:1963
1200:1961
1178:and
152:The
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