1157:
1007:
for low-dimensional manifolds?", meaning "If surgery worked in low dimensions, what would low-dimensional manifolds look like?" One can then compare the actual theory of low-dimensional manifolds to the low-dimensional analog of high-dimensional manifolds, and see if low-dimensional manifolds behave
999:
One can take a low-dimensional point of view on high-dimensional manifolds and ask "Which high-dimensional manifolds are geometrizable?", for various notions of geometrizable (cut into geometrizable pieces as in 3 dimensions, into symplectic manifolds, and so forth). In dimension 4 and above not all
984:
Thus dimension 4 differentiable manifolds are the most complicated: they are neither geometrizable (as in lower dimension), nor are they classified by surgery (as in higher dimension or topologically), and they exhibit unusual phenomena, most strikingly the uncountably infinitely many
652:, or more precisely, the triviality problem (given a finite presentation for a group, is it the trivial group?). Any finite presentation of a group can be realized as a 2-complex, and can be realized as the 2-skeleton of a 4-manifold (or higher). Thus one cannot even compute the
415:
Being homogeneous (away from any boundary), manifolds have no local point-set invariants, other than their dimension and boundary versus interior, and the most used global point-set properties are compactness and connectedness. Conventional names for combinations of these are:
1121:, 4-manifolds are classified by surgery, the differentiable classification question is phrased in terms of "differentiable structures": "which (topological) 4-manifolds admit a differentiable structure, and on those that do, how many differentiable structures are there?"
51:"Low dimensions" means dimensions up to 4; "high dimensions" means 5 or more dimensions. The case of dimension 4 is somehow a boundary case, as it manifests "low dimensional" behaviour smoothly (but not topologically); see
1022:
There is a unique connected 0-dimensional manifold, namely the point, and disconnected 0-dimensional manifolds are just discrete sets, classified by cardinality. They have no geometry, and their study is combinatorics.
1073:. There are 3 such curvatures (positive, zero, and negative). This is a classical result, and as stated, easy (the full uniformization theorem is subtler). The study of surfaces is deeply connected with
411:
The point-set classification is basicâone generally fixes point-set assumptions and then studies that class of manifold. The most frequently classified class of manifolds is closed, connected manifolds.
145:
These functors are in general neither one-to-one nor onto on objects; these failures are generally referred to in terms of "structure", as follows. A topological manifold that is in the image of
177:
140:
892:
846:
809:
355:, is little-used in the classification of manifolds, because these invariants are homotopy-invariant, and hence don't help with the finer classifications above homotopy type.
179:
is said to "admit a differentiable structure", and the fiber over a given topological manifold is "the different differentiable structures on the given topological manifold".
1030:, non-compact 1-dimensional manifold is homeomorphic or diffeomorphic to the real line. Dropping the assumption of second countability one gets two additional manifolds: the
729:
1008:"as you would expect": in what ways do they behave like high-dimensional manifolds (but for different reasons, or via different proofs) and in what ways are they unusual?
1403:
395:
1210:
934:
638:
256:
767:
693:
563:
531:
499:
60:
Different categories of manifolds yield different classifications; these are related by the notion of "structure", and more general categories have neater theories.
939:
This phenomenon is evident already for surfaces: there is a single orientable (and a single non-orientable) closed surface with positive curvature (the sphere and
1326:
In relative dimension, a submersion with compact domain is a fiber bundle (just as in codimension 0 = relative dimension 0), which are classified algebraically.
1364:
1114:
Four-dimensional manifolds are the most unusual: they are not geometrizable (as in lower dimensions), and surgery works topologically, but not differentiably.
472:
591:
and characteristic numbers are the corresponding generalized homological invariants, but they do not classify manifolds in higher dimension (they are not a
603:), so all characteristic classes vanish. In higher dimensions, characteristic classes do not in general vanish, and provide useful but not complete data.
1512:
Bull. Amer. Math. Soc. Volume 82, Number 5 (1976), 759-761; M. Kreck, Bordism of diffeomorphisms and related topics, Springer Lect. Notes 1069 (1984)
1481:
Proceedings of the international congress of mathematicians 2018, ICM 2018, Rio de
Janeiro, Brazil, August 1â9, 2018. Volume III. Invited lectures
2396:
659:
This ineffectiveness is a fundamental reason why surgery theory does not classify manifolds up to homeomorphism. Instead, for any fixed manifold
1587:
1258:
For maps, the appropriate notion of "low dimension" is for some purposes "self maps of low-dimensional manifolds", and for other purposes "low
108:: for instance, a differentiable manifold is also a topological manifold, and a differentiable map is also continuous, so there is a functor
2449:
2391:
217:. The most familiar example is orientability: some manifolds are orientable, some are not, and orientable manifolds admit 2 orientations.
97:. There are many different notions of "manifold", and corresponding notions of "map between manifolds", each of which yields a different
1678:
648:, there is no algorithm for determining if they are isomorphic (homeomorphic, diffeomorphic). This is due to the unsolvability of the
1702:
1897:
1026:
A connected compact 1-dimensional manifold without boundary is homeomorphic (or diffeomorphic if it is smooth) to the circle. A
1097:
1767:
1543:
424:
is a compact manifold, possibly with boundary, and not necessarily connected (but necessarily with finitely many components).
42:
Low-dimensional manifolds are classified by geometric structure; high-dimensional manifolds are classified algebraically, by
1993:
2046:
1574:
225:
There are two usual ways to give a classification: explicitly, by an enumeration, or implicitly, in terms of invariants.
2330:
1132:
993:
976:
High dimension manifolds (dimension 5 and more differentiably, dimension 4 and more topologically) are classified by
1164:
requires 2+1 dimensions (2 space, 1 time), hence the two
Whitney disks of surgery theory require 2+2+1=5 dimensions.
2095:
1124:
Four-manifolds often admit many unusual differentiable structures, most strikingly the uncountably infinitely many
214:
148:
111:
1003:
Conversely, one can take a high-dimensional point of view on low-dimensional manifolds and ask "What does surgery
903:
2078:
1687:
2439:
1339:
2290:
1697:
1044:
Maps from the circle into the 3-sphere (or more generally any 3-dimensional manifold) are studied as part of
851:
2275:
1998:
1772:
358:
Cobordism groups (the bordism groups of a point) are computed, but the bordism groups of a space (such as
352:
1335:
Particularly topologically interesting classes of maps include embeddings, immersions, and submersions.
2325:
2295:
2003:
1959:
1940:
1707:
1651:
592:
229:
1298:, they are rigid and geometric, and in the middle (codimension 2), one has a difficult exotic theory (
1294:
dimension (meaning more than 2), embeddings are classified by surgery, while in low codimension or in
1862:
1727:
1354:
1349:
1255:
of manifolds up to various equivalences, and there are many results and open questions in this area.
1093:
955:
814:
1309:
In codimension 2, particularly embeddings of 1-dimensional manifolds in 3-dimensional ones, one has
1089:. While the classification of surfaces is classical, maps of surfaces is an active area; see below.
702:
2247:
2112:
1804:
1646:
1247:, the classification of manifolds is one piece of understanding the category: it's classifying the
1069:
Every connected closed 2-dimensional manifold (surface) admits a constant curvature metric, by the
600:
1944:
1914:
1838:
1828:
1784:
1614:
1567:
1530:. Surveys on Surgery Theory (AM-145). Vol. 1. Princeton University Press. pp. 121â134.
1483:. World Scientific; Rio de Janeiro: Sociedade Brasileira de MatemĂĄtica (SBM). pp. 2035â2062.
1217:
1034:, and a space formed from a ray of the real line and a ray of the long line meeting at a point.
649:
84:
52:
1096:, and there are 8 such geometries. This is a recent result, and quite difficult. The proof (the
2285:
1904:
1799:
1712:
1619:
1359:
1070:
607:
578:
574:
361:
323:
206:
has more structure: in Diff it is simply a set, but in Top it is a group, and functorially so.
98:
67:
1495:
1479:
Navas, Andres (2018). "Group actions on 1-manifolds: a list of very concrete open questions".
1183:
1934:
1929:
1423:
1323:, which are classified algebraically, and these are more naturally thought of as submersions.
913:
772:
617:
235:
21:
2265:
2203:
2051:
1755:
1745:
1717:
1692:
1602:
1383:
1232:
1031:
740:
733:
666:
588:
536:
504:
477:
333:
296:
263:
1316:
In codimension 1, a codimension 1 embedding separates a manifold, and these are tractable.
1092:
Every closed 3-dimensional manifold can be cut into pieces that are geometrizable, by the
8:
2403:
2085:
1963:
1948:
1877:
1636:
1275:
259:
2376:
2444:
2345:
2300:
2197:
2068:
1872:
1560:
1375:
1295:
1149:
In dimension 5 and above (and 4 dimensions topologically), manifolds are classified by
1078:
1058:
313:
291:
273:
1882:
445:
438:
is a manifold without boundary (not necessarily connected), with no compact component.
2280:
2260:
2255:
2162:
2073:
1887:
1867:
1722:
1661:
1539:
910:. Conversely, negative curvature is generic: for instance, any manifold of dimension
653:
301:
105:
2418:
2212:
2167:
2090:
2061:
1919:
1852:
1847:
1842:
1832:
1624:
1607:
1531:
1461:
1177:
1074:
1037:
The study of maps of 1-dimensional manifolds are a non-trivial area. For example:
1027:
940:
906:
show that manifolds with positive curvature are constrained, most dramatically the
897:
1509:
1231:, but when the codimension is more than 2, embedding theory is tractable, via the
2361:
2270:
2100:
2056:
1822:
1244:
1086:
1082:
1041:
Groups of diffeomorphisms of 1-manifolds are quite difficult to understand finely
973:
Middle dimension manifolds (dimension 4 differentiably) exhibit exotic phenomena.
406:
329:
306:
283:
28:
is a basic question, about which much is known, and many open questions remain.
2227:
2152:
2122:
2020:
2013:
1953:
1924:
1794:
1789:
1750:
1320:
1150:
1144:
1131:. Similarly, differentiable 4-manifolds is the only remaining open case of the
977:
907:
596:
339:
43:
1535:
1465:
2433:
2413:
2237:
2232:
2217:
2207:
2157:
2134:
2008:
1968:
1909:
1857:
1656:
1279:
1213:
1169:
1161:
992:. Notably, differentiable 4-manifolds is the only remaining open case of the
319:
278:
1449:
2340:
2335:
2177:
2144:
2117:
2025:
1666:
1408:
1306:
In codimension greater than 2, embeddings are classified by surgery theory.
1173:
951:), and all surfaces of higher genus admit negative curvature metrics only.
948:
74:, for instance), there is no algorithm to determine if they are isomorphic.
1212:). In dimension 4, one can resolve intersections of two Whitney disks via
198:
More precisely, what is the structure of the set of additional structures?
2183:
2172:
2129:
2030:
1631:
1310:
1299:
1259:
1228:
1224:
1223:
More subtly, dimension 5 is the cut-off because the middle dimension has
1045:
210:
94:
17:
2408:
2366:
2192:
2105:
1737:
1641:
1552:
1414:
Diffeomorphisms up to cobordism have been classified by
Matthias Kreck
1109:
1064:
645:
641:
582:
71:
1526:
Kreck, Matthias (2000). "A guide to the classification of manifolds".
2222:
2187:
1892:
1779:
1398:
1125:
986:
348:
1156:
2386:
2381:
2371:
1762:
1583:
1427:
1393:
90:
656:
of a given high-dimensional manifold, much less a classification.
431:
is a compact manifold without boundary, not necessarily connected.
63:
Positive curvature is constrained, negative curvature is generic.
1000:
manifolds are geometrizable, but they are an interesting class.
898:
Positive curvature is constrained, negative curvature is generic
78:
1978:
1499:. Netherlands, Springer Netherlands, 1991. 333.
811:, being regarded as equivalent if there exist a homeomorphism
1017:
944:
193:
If it admits an additional structure, how many does it admit?
66:
The abstract classification of high-dimensional manifolds is
970:
Low dimension manifolds (dimensions 2 and 3) admit geometry.
1172:
works in the middle dimension in dimension 5 and more: two
1345:
Fundamental results in embeddings and immersions include:
1319:
In codimension 0, a codimension 0 (proper) immersion is a
182:
Thus given two categories, the two natural questions are:
1176:
generically don't intersect in dimension 5 and above, by
1290:
Analogously to the classification of manifolds, in high
1216:, which works topologically but not differentiably; see
1227:
more than 2: when the codimension is 2, one encounters
163:
153:
126:
116:
1186:
916:
854:
817:
775:
743:
705:
669:
620:
539:
507:
480:
448:
364:
238:
151:
114:
1496:
Discrete groups in space and uniformization problems
1052:
269:
Manifolds have a rich set of invariants, including:
1454:
258:tori, and an invariant that classifies them is the
1450:"On the classification of 1-dimensional manifolds"
1389:One may classify maps up to various equivalences:
1204:
1081:, as every orientable surface can be considered a
928:
886:
840:
803:
761:
723:
687:
632:
557:
525:
493:
466:
389:
250:
171:
134:
2431:
1138:
967:Dimension 0 is trivial and 1 is straightforward.
220:
936:admits a metric with negative Ricci curvature.
585:, so 2-manifolds are classified homologically.
1568:
1528:Princeton University Press eBook Package 2014
606:Manifolds in dimension 4 and above cannot be
172:{\displaystyle {\mbox{Diff}}\to {\mbox{Top}}}
135:{\displaystyle {\mbox{Diff}}\to {\mbox{Top}}}
79:Different categories and additional structure
1265:
958:, all but hyperbolic are quite constrained.
595:): for instance, orientable 3-manifolds are
1271:1-dimensional: homeomorphisms of the circle
228:For instance, for orientable surfaces, the
53:discussion of "low" versus "high" dimension
1575:
1561:
1238:
904:classical theorems in Riemannian geometry
101:and a different classification question.
1582:
1155:
961:
943:), and likewise for zero curvature (the
232:enumerates them as the connected sum of
1168:The reason for dimension 5 is that the
2432:
1447:
1370:Key tools in studying these maps are:
1103:
887:{\displaystyle f'h\sim f\colon N\to M}
1556:
1525:
1478:
1441:
1011:
1502:
1251:. The other question is classifying
1126:exotic differentiable structures on
987:exotic differentiable structures on
70:: given two manifolds (presented as
2450:Mathematical classification systems
1235:. This is discussed further below.
1098:Solution of the Poincaré conjecture
13:
1519:
1330:
1285:
954:Similarly for 3-manifolds: of the
347:Modern algebraic topology (beyond
14:
2461:
1053:Dimensions 2 and 3: geometrizable
202:In more general categories, this
1100:) is analytic, not topological.
568:
215:reduction of the structure group
186:Which manifolds of a given type
104:These categories are related by
1133:generalized Poincaré conjecture
994:generalized Poincaré conjecture
841:{\displaystyle h\colon N\to N'}
1615:Differentiable/Smooth manifold
1487:
1472:
1338:Geometrically interesting are
878:
827:
798:
776:
756:
744:
724:{\displaystyle f\colon N\to M}
715:
682:
670:
573:The Euler characteristic is a
552:
540:
520:
508:
461:
449:
384:
378:
159:
122:
31:
1:
1434:
1218:Geometric topology: Dimension
1139:Dimension 5 and more: surgery
326:, and characteristic numbers)
221:Enumeration versus invariants
209:Many of these structures are
93:is classifying objects up to
400:
7:
2321:Classification of manifolds
1417:
577:invariant, and thus can be
533:is an open manifold, while
36:
26:classification of manifolds
10:
2466:
1510:Bordism of diffeomorphisms
1342:and isometric immersions.
1243:From the point of view of
1220:for details on dimension.
1142:
1107:
1062:
1056:
1015:
908:1/4-pinched sphere theorem
593:complete set of invariants
501:is a closed manifold, and
404:
353:Extraordinary (co)homology
230:classification of surfaces
82:
2397:over commutative algebras
2354:
2313:
2246:
2143:
2039:
1986:
1977:
1813:
1736:
1675:
1595:
1536:10.1515/9781400865192-009
1424:The Berger classification
1355:Whitney immersion theorem
1350:Whitney embedding theorem
1266:Low-dimensional self-maps
1094:geometrization conjecture
390:{\displaystyle MO_{*}(M)}
2113:Riemann curvature tensor
1493:Apanasov, B..
1205:{\displaystyle 2+2<5}
601:low-dimensional topology
190:an additional structure?
1448:FrolĂk, ZdenÄk (1962).
929:{\displaystyle n\geq 3}
804:{\displaystyle (N',f')}
650:word problem for groups
633:{\displaystyle n\geq 4}
599:(Steenrod's theorem in
474:is a compact manifold,
251:{\displaystyle n\geq 0}
85:Categories of manifolds
1905:Manifold with boundary
1620:Differential structure
1360:Nash embedding theorem
1239:Maps between manifolds
1206:
1165:
1071:uniformization theorem
930:
888:
842:
805:
763:
725:
689:
634:
610:classified: given two
589:Characteristic classes
559:
527:
495:
468:
391:
324:characteristic classes
252:
213:, and the question is
173:
136:
89:Formally, classifying
2440:Differential geometry
1207:
1159:
1143:Further information:
1108:Further information:
1063:Further information:
1057:Further information:
1016:Further information:
962:Overview by dimension
931:
889:
843:
806:
764:
762:{\displaystyle (N,f)}
726:
690:
688:{\displaystyle (N,f)}
635:
560:
558:{\displaystyle [0,1)}
528:
526:{\displaystyle (0,1)}
496:
494:{\displaystyle S^{1}}
469:
405:Further information:
397:) are generally not.
392:
253:
174:
137:
83:Further information:
22:geometry and topology
2052:Covariant derivative
1603:Topological manifold
1384:Calculus of functors
1365:Smale-Hirsch theorem
1233:calculus of functors
1184:
914:
852:
815:
773:
741:
734:homotopy equivalence
703:
667:
663:it classifies pairs
618:
579:effectively computed
537:
505:
478:
446:
362:
334:Reidemeister torsion
297:Euler characteristic
264:Euler characteristic
236:
149:
112:
2086:Exterior derivative
1688:AtiyahâSinger index
1637:Riemannian manifold
1276:mapping class group
1104:Dimension 4: exotic
318:normal invariants (
2392:Secondary calculus
2346:Singularity theory
2301:Parallel transport
2069:De Rham cohomology
1708:Generalized Stokes
1466:10338.dmlcz/142137
1296:relative dimension
1202:
1166:
1079:algebraic geometry
1059:Surface (topology)
1012:Dimensions 0 and 1
926:
884:
838:
801:
759:
737:, two such pairs,
721:
685:
630:
565:is none of these.
555:
523:
491:
464:
387:
314:Geometric topology
292:algebraic topology
274:Point-set topology
248:
169:
167:
157:
132:
130:
120:
106:forgetful functors
2427:
2426:
2309:
2308:
2074:Differential form
1728:Whitney embedding
1662:Differential form
1545:978-1-4008-6519-2
654:fundamental group
351:theory), such as
302:Fundamental group
166:
156:
129:
119:
2457:
2419:Stratified space
2377:Fréchet manifold
2091:Interior product
1984:
1983:
1681:
1577:
1570:
1563:
1554:
1553:
1549:
1513:
1506:
1500:
1491:
1485:
1484:
1476:
1470:
1469:
1445:
1211:
1209:
1208:
1203:
1178:general position
1075:complex analysis
1028:second countable
941:projective plane
935:
933:
932:
927:
893:
891:
890:
885:
862:
847:
845:
844:
839:
837:
810:
808:
807:
802:
797:
786:
768:
766:
765:
760:
730:
728:
727:
722:
694:
692:
691:
686:
639:
637:
636:
631:
564:
562:
561:
556:
532:
530:
529:
524:
500:
498:
497:
492:
490:
489:
473:
471:
470:
467:{\displaystyle }
465:
422:compact manifold
396:
394:
393:
388:
377:
376:
257:
255:
254:
249:
178:
176:
175:
170:
168:
164:
158:
154:
141:
139:
138:
133:
131:
127:
121:
117:
2465:
2464:
2460:
2459:
2458:
2456:
2455:
2454:
2430:
2429:
2428:
2423:
2362:Banach manifold
2355:Generalizations
2350:
2305:
2242:
2139:
2101:Ricci curvature
2057:Cotangent space
2035:
1973:
1815:
1809:
1768:Exponential map
1732:
1677:
1671:
1591:
1581:
1546:
1522:
1520:Further reading
1517:
1516:
1507:
1503:
1492:
1488:
1477:
1473:
1446:
1442:
1437:
1420:
1333:
1331:High dimensions
1288:
1286:Low codimension
1274:2-dimensional:
1268:
1245:category theory
1241:
1185:
1182:
1181:
1147:
1141:
1112:
1106:
1087:algebraic curve
1083:Riemann surface
1067:
1061:
1055:
1020:
1014:
964:
915:
912:
911:
900:
855:
853:
850:
849:
848:and a homotopy
830:
816:
813:
812:
790:
779:
774:
771:
770:
742:
739:
738:
704:
701:
700:
699:a manifold and
668:
665:
664:
640:) presented as
619:
616:
615:
571:
538:
535:
534:
506:
503:
502:
485:
481:
479:
476:
475:
447:
444:
443:
429:closed manifold
409:
407:closed manifold
403:
372:
368:
363:
360:
359:
330:Simple homotopy
307:Cohomology ring
237:
234:
233:
223:
162:
152:
150:
147:
146:
125:
115:
113:
110:
109:
87:
81:
39:
34:
20:, specifically
12:
11:
5:
2463:
2453:
2452:
2447:
2442:
2425:
2424:
2422:
2421:
2416:
2411:
2406:
2401:
2400:
2399:
2389:
2384:
2379:
2374:
2369:
2364:
2358:
2356:
2352:
2351:
2349:
2348:
2343:
2338:
2333:
2328:
2323:
2317:
2315:
2311:
2310:
2307:
2306:
2304:
2303:
2298:
2293:
2288:
2283:
2278:
2273:
2268:
2263:
2258:
2252:
2250:
2244:
2243:
2241:
2240:
2235:
2230:
2225:
2220:
2215:
2210:
2200:
2195:
2190:
2180:
2175:
2170:
2165:
2160:
2155:
2149:
2147:
2141:
2140:
2138:
2137:
2132:
2127:
2126:
2125:
2115:
2110:
2109:
2108:
2098:
2093:
2088:
2083:
2082:
2081:
2071:
2066:
2065:
2064:
2054:
2049:
2043:
2041:
2037:
2036:
2034:
2033:
2028:
2023:
2018:
2017:
2016:
2006:
2001:
1996:
1990:
1988:
1981:
1975:
1974:
1972:
1971:
1966:
1956:
1951:
1937:
1932:
1927:
1922:
1917:
1915:Parallelizable
1912:
1907:
1902:
1901:
1900:
1890:
1885:
1880:
1875:
1870:
1865:
1860:
1855:
1850:
1845:
1835:
1825:
1819:
1817:
1811:
1810:
1808:
1807:
1802:
1797:
1795:Lie derivative
1792:
1790:Integral curve
1787:
1782:
1777:
1776:
1775:
1765:
1760:
1759:
1758:
1751:Diffeomorphism
1748:
1742:
1740:
1734:
1733:
1731:
1730:
1725:
1720:
1715:
1710:
1705:
1700:
1695:
1690:
1684:
1682:
1673:
1672:
1670:
1669:
1664:
1659:
1654:
1649:
1644:
1639:
1634:
1629:
1628:
1627:
1622:
1612:
1611:
1610:
1599:
1597:
1596:Basic concepts
1593:
1592:
1580:
1579:
1572:
1565:
1557:
1551:
1550:
1544:
1521:
1518:
1515:
1514:
1501:
1486:
1471:
1439:
1438:
1436:
1433:
1432:
1431:
1419:
1416:
1412:
1411:
1406:
1401:
1396:
1387:
1386:
1381:
1368:
1367:
1362:
1357:
1352:
1332:
1329:
1328:
1327:
1324:
1321:covering space
1317:
1314:
1307:
1287:
1284:
1283:
1282:
1272:
1267:
1264:
1240:
1237:
1214:Casson handles
1201:
1198:
1195:
1192:
1189:
1151:surgery theory
1145:surgery theory
1140:
1137:
1105:
1102:
1054:
1051:
1050:
1049:
1042:
1013:
1010:
982:
981:
978:surgery theory
974:
971:
968:
963:
960:
925:
922:
919:
899:
896:
883:
880:
877:
874:
871:
868:
865:
861:
858:
836:
833:
829:
826:
823:
820:
800:
796:
793:
789:
785:
782:
778:
758:
755:
752:
749:
746:
720:
717:
714:
711:
708:
684:
681:
678:
675:
672:
629:
626:
623:
597:parallelizable
570:
567:
554:
551:
548:
545:
542:
522:
519:
516:
513:
510:
488:
484:
463:
460:
457:
454:
451:
442:For instance,
440:
439:
432:
425:
402:
399:
386:
383:
380:
375:
371:
367:
345:
344:
343:
342:
340:Surgery theory
337:
327:
311:
310:
309:
304:
299:
288:
287:
286:
281:
247:
244:
241:
222:
219:
200:
199:
195:
194:
191:
161:
124:
80:
77:
76:
75:
64:
61:
57:
56:
48:
47:
44:surgery theory
38:
35:
33:
30:
9:
6:
4:
3:
2:
2462:
2451:
2448:
2446:
2443:
2441:
2438:
2437:
2435:
2420:
2417:
2415:
2414:Supermanifold
2412:
2410:
2407:
2405:
2402:
2398:
2395:
2394:
2393:
2390:
2388:
2385:
2383:
2380:
2378:
2375:
2373:
2370:
2368:
2365:
2363:
2360:
2359:
2357:
2353:
2347:
2344:
2342:
2339:
2337:
2334:
2332:
2329:
2327:
2324:
2322:
2319:
2318:
2316:
2312:
2302:
2299:
2297:
2294:
2292:
2289:
2287:
2284:
2282:
2279:
2277:
2274:
2272:
2269:
2267:
2264:
2262:
2259:
2257:
2254:
2253:
2251:
2249:
2245:
2239:
2236:
2234:
2231:
2229:
2226:
2224:
2221:
2219:
2216:
2214:
2211:
2209:
2205:
2201:
2199:
2196:
2194:
2191:
2189:
2185:
2181:
2179:
2176:
2174:
2171:
2169:
2166:
2164:
2161:
2159:
2156:
2154:
2151:
2150:
2148:
2146:
2142:
2136:
2135:Wedge product
2133:
2131:
2128:
2124:
2121:
2120:
2119:
2116:
2114:
2111:
2107:
2104:
2103:
2102:
2099:
2097:
2094:
2092:
2089:
2087:
2084:
2080:
2079:Vector-valued
2077:
2076:
2075:
2072:
2070:
2067:
2063:
2060:
2059:
2058:
2055:
2053:
2050:
2048:
2045:
2044:
2042:
2038:
2032:
2029:
2027:
2024:
2022:
2019:
2015:
2012:
2011:
2010:
2009:Tangent space
2007:
2005:
2002:
2000:
1997:
1995:
1992:
1991:
1989:
1985:
1982:
1980:
1976:
1970:
1967:
1965:
1961:
1957:
1955:
1952:
1950:
1946:
1942:
1938:
1936:
1933:
1931:
1928:
1926:
1923:
1921:
1918:
1916:
1913:
1911:
1908:
1906:
1903:
1899:
1896:
1895:
1894:
1891:
1889:
1886:
1884:
1881:
1879:
1876:
1874:
1871:
1869:
1866:
1864:
1861:
1859:
1856:
1854:
1851:
1849:
1846:
1844:
1840:
1836:
1834:
1830:
1826:
1824:
1821:
1820:
1818:
1812:
1806:
1803:
1801:
1798:
1796:
1793:
1791:
1788:
1786:
1783:
1781:
1778:
1774:
1773:in Lie theory
1771:
1770:
1769:
1766:
1764:
1761:
1757:
1754:
1753:
1752:
1749:
1747:
1744:
1743:
1741:
1739:
1735:
1729:
1726:
1724:
1721:
1719:
1716:
1714:
1711:
1709:
1706:
1704:
1701:
1699:
1696:
1694:
1691:
1689:
1686:
1685:
1683:
1680:
1676:Main results
1674:
1668:
1665:
1663:
1660:
1658:
1657:Tangent space
1655:
1653:
1650:
1648:
1645:
1643:
1640:
1638:
1635:
1633:
1630:
1626:
1623:
1621:
1618:
1617:
1616:
1613:
1609:
1606:
1605:
1604:
1601:
1600:
1598:
1594:
1589:
1585:
1578:
1573:
1571:
1566:
1564:
1559:
1558:
1555:
1547:
1541:
1537:
1533:
1529:
1524:
1523:
1511:
1505:
1498:
1497:
1490:
1482:
1475:
1467:
1463:
1459:
1455:
1451:
1444:
1440:
1429:
1425:
1422:
1421:
1415:
1410:
1407:
1405:
1402:
1400:
1397:
1395:
1392:
1391:
1390:
1385:
1382:
1380:
1378:
1373:
1372:
1371:
1366:
1363:
1361:
1358:
1356:
1353:
1351:
1348:
1347:
1346:
1343:
1341:
1336:
1325:
1322:
1318:
1315:
1312:
1308:
1305:
1304:
1303:
1301:
1297:
1293:
1281:
1280:Torelli group
1277:
1273:
1270:
1269:
1263:
1261:
1256:
1254:
1250:
1246:
1236:
1234:
1230:
1226:
1221:
1219:
1215:
1199:
1196:
1193:
1190:
1187:
1179:
1175:
1174:Whitney disks
1171:
1170:Whitney trick
1163:
1162:Whitney trick
1158:
1154:
1152:
1146:
1136:
1134:
1130:
1129:
1122:
1120:
1119:topologically
1115:
1111:
1101:
1099:
1095:
1090:
1088:
1084:
1080:
1076:
1072:
1066:
1060:
1047:
1043:
1040:
1039:
1038:
1035:
1033:
1029:
1024:
1019:
1009:
1006:
1001:
997:
995:
991:
990:
979:
975:
972:
969:
966:
965:
959:
957:
952:
950:
946:
942:
937:
923:
920:
917:
909:
905:
895:
881:
875:
872:
869:
866:
863:
859:
856:
834:
831:
824:
821:
818:
794:
791:
787:
783:
780:
753:
750:
747:
736:
735:
718:
712:
709:
706:
698:
679:
676:
673:
662:
657:
655:
651:
647:
643:
627:
624:
621:
613:
609:
604:
602:
598:
594:
590:
586:
584:
580:
576:
569:Computability
566:
549:
546:
543:
517:
514:
511:
486:
482:
458:
455:
452:
437:
436:open manifold
433:
430:
426:
423:
419:
418:
417:
413:
408:
398:
381:
373:
369:
365:
356:
354:
350:
341:
338:
335:
331:
328:
325:
321:
320:orientability
317:
316:
315:
312:
308:
305:
303:
300:
298:
295:
294:
293:
289:
285:
284:Connectedness
282:
280:
277:
276:
275:
272:
271:
270:
267:
265:
261:
245:
242:
239:
231:
226:
218:
216:
212:
207:
205:
204:structure set
197:
196:
192:
189:
185:
184:
183:
180:
143:
107:
102:
100:
96:
92:
86:
73:
69:
65:
62:
59:
58:
54:
50:
49:
45:
41:
40:
29:
27:
23:
19:
2341:Moving frame
2336:Morse theory
2326:Gauge theory
2320:
2118:Tensor field
2047:Closed/Exact
2026:Vector field
1994:Distribution
1935:Hypercomplex
1930:Quaternionic
1667:Vector field
1625:Smooth atlas
1527:
1504:
1494:
1489:
1480:
1474:
1457:
1453:
1443:
1413:
1388:
1376:
1369:
1344:
1337:
1334:
1291:
1289:
1257:
1252:
1248:
1242:
1222:
1167:
1148:
1127:
1123:
1118:
1116:
1113:
1091:
1068:
1036:
1025:
1021:
1004:
1002:
998:
988:
983:
956:8 geometries
953:
949:Klein bottle
938:
901:
732:
696:
660:
658:
646:handlebodies
642:CW complexes
614:-manifolds (
611:
605:
587:
583:CW structure
572:
441:
435:
428:
421:
414:
410:
357:
346:
268:
227:
224:
211:G-structures
208:
203:
201:
187:
181:
144:
103:
88:
72:CW complexes
25:
15:
2286:Levi-Civita
2276:Generalized
2248:Connections
2198:Lie algebra
2130:Volume form
2031:Vector flow
2004:Pushforward
1999:Lie bracket
1898:Lie algebra
1863:G-structure
1652:Pushforward
1632:Submanifold
1404:concordance
1379:-principles
1311:knot theory
1300:knot theory
1260:codimension
1229:knot theory
1225:codimension
1085:or complex
1046:knot theory
608:effectively
575:homological
279:Compactness
95:isomorphism
68:ineffective
32:Main themes
18:mathematics
2434:Categories
2409:Stratifold
2367:Diffeology
2163:Associated
1964:Symplectic
1949:Riemannian
1878:Hyperbolic
1805:Submersion
1713:HopfâRinow
1647:Submersion
1642:Smooth map
1508:M. Kreck,
1460:(1): 1â4.
1435:References
1340:isometries
1110:4-manifold
1065:3-manifold
2445:Manifolds
2291:Principal
2266:Ehresmann
2223:Subbundle
2213:Principal
2188:Fibration
2168:Cotangent
2040:Covectors
1893:Lie group
1873:Hermitian
1816:manifolds
1785:Immersion
1780:Foliation
1718:Noether's
1703:Frobenius
1698:De Rham's
1693:Darboux's
1584:Manifolds
1399:cobordism
1374:Gromov's
1032:long line
921:≥
879:→
873::
867:∼
828:→
822::
716:→
710::
625:≥
401:Point-set
374:∗
349:cobordism
243:≥
160:→
123:→
91:manifolds
2387:Orbifold
2382:K-theory
2372:Diffiety
2096:Pullback
1910:Oriented
1888:Kenmotsu
1868:Hadamard
1814:Types of
1763:Geodesic
1588:Glossary
1428:holonomy
1418:See also
1394:homotopy
947:and the
860:′
835:′
795:′
784:′
581:given a
290:Classic
99:category
37:Overview
2331:History
2314:Related
2228:Tangent
2206:)
2186:)
2153:Adjoint
2145:Bundles
2123:density
2021:Torsion
1987:Vectors
1979:Tensors
1962:)
1947:)
1943:,
1941:Pseudoâ
1920:Poisson
1853:Finsler
1848:Fibered
1843:Contact
1841:)
1833:Complex
1831:)
1800:Section
1430:groups.
1409:isotopy
1249:objects
1005:predict
2296:Vector
2281:Koszul
2261:Cartan
2256:Affine
2238:Vector
2233:Tensor
2218:Spinor
2208:Normal
2204:Stable
2158:Affine
2062:bundle
2014:bundle
1960:Almost
1883:KĂ€hler
1839:Almost
1829:Almost
1823:Closed
1723:Sard's
1679:(list)
1542:
1117:Since
24:, the
2404:Sheaf
2178:Fiber
1954:Rizza
1925:Prime
1756:Local
1746:Curve
1608:Atlas
1018:Curve
945:torus
902:Many
695:with
260:genus
188:admit
2271:Form
2173:Dual
2106:flow
1969:Tame
1945:Subâ
1858:Flat
1738:Maps
1540:ISBN
1278:and
1253:maps
1197:<
1160:The
1077:and
769:and
155:Diff
118:Diff
2193:Jet
1532:doi
1462:hdl
1426:of
1302:).
1262:".
644:or
434:An
262:or
165:Top
128:Top
16:In
2436::
2184:Co
1538:.
1456:.
1452:.
1292:co
1153:.
1135:.
996:.
894:.
731:a
427:A
420:A
322:,
266:.
142:.
2202:(
2182:(
1958:(
1939:(
1837:(
1827:(
1590:)
1586:(
1576:e
1569:t
1562:v
1548:.
1534::
1468:.
1464::
1458:3
1377:h
1313:.
1200:5
1194:2
1191:+
1188:2
1180:(
1128:R
1048:.
989:R
980:.
924:3
918:n
882:M
876:N
870:f
864:h
857:f
832:N
825:N
819:h
799:)
792:f
788:,
781:N
777:(
757:)
754:f
751:,
748:N
745:(
719:M
713:N
707:f
697:N
683:)
680:f
677:,
674:N
671:(
661:M
628:4
622:n
612:n
553:)
550:1
547:,
544:0
541:[
521:)
518:1
515:,
512:0
509:(
487:1
483:S
462:]
459:1
456:,
453:0
450:[
385:)
382:M
379:(
370:O
366:M
336:)
332:(
246:0
240:n
55:.
46:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.