1101:. There then arises the question whether orientation-reversing diffeomorphisms exist. There is an "essentially unique" smooth structure for any topological manifold of dimension smaller than 4. For compact manifolds of dimension greater than 4, there is a finite number of "smooth types", i.e. equivalence classes of pairwise smoothly diffeomorphic smooth structures. In the case of
1356:
proved that the number of smooth structures on a compact PL manifold is finite and agrees with the number of differential structures on the sphere for the same dimension (see the book
Asselmeyer-Maluga, Brans chapter 7) . By combining these results, the number of smooth structures on a compact
1534:
757:
1385:
4-manifold, one can use a surgery along a knot or link to produce a new differential structure. With the help of this procedure one can produce countably infinite many differential structures. But even for simple spaces such as
226:
1062:âdiffeomorphism identifying the two. It follows that there is only one class of smooth structures (modulo pairwise smooth diffeomorphism) over any topological manifold which admits a differentiable structure, i.e. The
1007:âmanifold. This maximal atlas then uniquely determines both the topology and the underlying set, the latter being the union of the domains of all charts, and the former having the set of all these domains as a basis.
590:
522:
662:
387:
330:
1443:
911:
801:
449:
268:
149:
858:
1448:
668:
157:
1295:
has, except that there is at least one. There may be one, a finite number, or an infinite number. The claim that there is just one is known as the
996:. Each distinct differential structure is determined by a unique maximal atlas, which is simply the union of all atlases in the equivalence class.
2555:
1746:
1312:
has more than one smooth type. The problem is connected with the existence of more than one smooth type of the topological 4-disk (or 4-ball).
1320:
As mentioned above, in dimensions smaller than 4, there is only one differential structure for each topological manifold. That was proved by
2550:
1837:
1861:
2056:
1926:
1445:
one doesn't know the construction of other differential structures. For non-compact 4-manifolds there are many examples like
528:
460:
605:
2152:
1070:âmanifold. A bit loosely, one might express this by saying that the smooth structure is (essentially) unique. The case for
336:
279:
2205:
1733:
28:
1363:
is more complicated. For compact manifolds, results depend on the complexity of the manifold as measured by the second
2489:
1304:
1699:
1604:
1586:
2598:
2254:
1389:
869:
2237:
1846:
765:
2449:
1856:
398:
2434:
2157:
1931:
69:
is already a topological manifold, it is required that the new topology be identical to the existing one.
2479:
2484:
2454:
2162:
2118:
2099:
1866:
1810:
1091:
244:
125:
2021:
1886:
1097:
Smooth structures on an orientable manifold are usually counted modulo orientation-preserving smooth
816:
2406:
2271:
1963:
1805:
2103:
2073:
1997:
1987:
1943:
1773:
1726:
2444:
2063:
1958:
1871:
1545:
1087:
1299:
270:, but the usefulness of this depends on how much the charts agree when their domains overlap.
2093:
2088:
1683:
1636:
1634:(1952). "Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung".
1337:
999:
For simplification of language, without any loss of precision, one might just call a maximal
62:
54:
2424:
2362:
2210:
1914:
1904:
1876:
1851:
1761:
1665:
1529:{\displaystyle {\mathbb {R} }^{4},S^{3}\times {\mathbb {R} },M^{4}\smallsetminus \{*\},...}
1075:
58:
752:{\displaystyle \varphi _{ij}(x)=\varphi _{j}\left(\varphi _{i}^{-1}\left(x\right)\right).}
8:
2562:
2244:
2122:
2107:
2036:
1795:
981:
2535:
927: = 0, we only require that the transition maps are continuous, consequently a
2504:
2459:
2356:
2227:
2031:
1719:
1688:
1653:
1329:
2041:
2439:
2419:
2414:
2321:
2232:
2046:
2026:
1881:
1820:
1695:
1582:
39:
221:{\displaystyle \varphi _{i}:M\supset W_{i}\rightarrow U_{i}\subset \mathbb {R} ^{n}}
2577:
2371:
2326:
2249:
2220:
2078:
2011:
2006:
2001:
1991:
1783:
1766:
1645:
1613:
1382:
102:
2520:
2429:
2259:
2215:
1981:
1661:
1599:
1349:
1150:âdifferential structure not smoothly diffeomorphic to the usual one are known as
1032:
2386:
2311:
2281:
2179:
2172:
2112:
2083:
1953:
1948:
1909:
1631:
1325:
599:
between the two charts translates between their images on their shared domain:
596:
2592:
2572:
2396:
2391:
2376:
2366:
2316:
2293:
2167:
2127:
2068:
2016:
1815:
1574:
1555:
1353:
1151:
1129:
1098:
241:
Each chart allows a subset of the manifold to be viewed as an open subset of
1690:
Foundational Essays on
Topological Manifolds. Smoothings, and Triangulations
1321:
2499:
2494:
2336:
2303:
2276:
2184:
1825:
1679:
1364:
1341:
1333:
1123:
935: = â, derivatives of all orders must be continuous. A family of
2342:
2331:
2288:
2189:
1790:
1345:
1344:
for compact topological manifolds of dimension greater than 4 is finite.
1291:
It is not currently known how many smooth types the topological 4-sphere
20:
1602:(1960). "A manifold which does not admit any differentiable structure".
1308:). Most mathematicians believe that this conjecture is false, i.e. that
1134:
The following table lists the number of smooth types of the topological
2567:
2525:
2351:
2264:
1896:
1800:
1711:
1657:
1617:
1360:
917:
107:
2381:
2346:
2051:
1938:
1550:
1114:
1649:
2545:
2540:
2530:
1921:
1742:
1315:
1113:= 4, there are uncountably many such types. One refers to these by
993:
970:-atlas that defines a topological manifold is said to determine a
931:-atlas is simply another way to define a topological manifold. If
2137:
1589:. for a general mathematical account of differential structures
1357:
topological manifold of dimension not equal to 4 is finite.
1124:
Differential structures on spheres of dimension 1 to 20
585:{\displaystyle U_{ji}=\varphi _{j}\left(W_{ij}\right).}
517:{\displaystyle U_{ij}=\varphi _{i}\left(W_{ij}\right),}
1031:âatlas on the same underlying set by a theorem due to
657:{\displaystyle \varphi _{ij}:U_{ij}\rightarrow U_{ji}}
1694:. Princeton, New Jersey: Princeton University Press.
1451:
1392:
939:-compatible charts covering the whole manifold is a
872:
819:
768:
671:
608:
531:
463:
401:
339:
282:
247:
160:
128:
1010:
382:{\displaystyle \varphi _{j}:W_{j}\rightarrow U_{j}.}
325:{\displaystyle \varphi _{i}:W_{i}\rightarrow U_{i},}
85:
which may be a non-negative integer or infinity, an
1109:â 4, the number of these types is one, whereas for
1687:
1528:
1437:
905:
852:
795:
751:
656:
584:
516:
443:
381:
324:
262:
220:
143:
1536:having uncountably many differential structures.
2590:
1316:Differential structures on topological manifolds
1678:
61:with some additional structure that allows for
1727:
1438:{\displaystyle S^{4},{\mathbb {C} }P^{2},...}
1146:from 1 up to 20. Spheres with a smooth, i.e.
954:if the union of their sets of charts forms a
906:{\displaystyle \varphi _{ij},\,\varphi _{ji}}
1511:
1505:
796:{\displaystyle \varphi _{i},\,\varphi _{j}}
1734:
1720:
1035:. It has also been shown that any maximal
1484:
1455:
1408:
889:
836:
782:
250:
208:
131:
1741:
1598:
1086:, and later explained in the context of
1083:
1027:âmanifold, the maximal atlas contains a
918:continuous partial derivatives of order
1051:> 0, although for any pair of these
947:differential manifold. Two atlases are
454:whose images under the two charts are
2591:
1074:= 0 is different. Namely, there exist
444:{\displaystyle W_{ij}=W_{i}\cap W_{j}}
1715:
1630:
1340:were able to show that the number of
392:The intersection of their domains is
13:
863:are open, and the transition maps
14:
2610:
1605:Commentarii Mathematici Helvetici
1011:Existence and uniqueness theorems
977:on the topological manifold. The
1142:for the values of the dimension
263:{\displaystyle \mathbb {R} ^{n}}
144:{\displaystyle \mathbb {R} ^{n}}
1305:Generalized Poincaré conjecture
1082:âstructure, a result proved by
1039:âatlas contains some number of
853:{\displaystyle U_{ij},\,U_{ji}}
1774:Differentiable/Smooth manifold
1672:
1624:
1592:
1568:
1324:for dimension 1 and 2, and by
691:
685:
638:
363:
306:
238:(in the sense defined below):
190:
1:
1561:
118:(whose union is the whole of
72:
7:
2480:Classification of manifolds
1539:
10:
2615:
1374:. For large Betti numbers
1127:
2556:over commutative algebras
2513:
2472:
2405:
2302:
2198:
2145:
2136:
1972:
1895:
1834:
1754:
1328:in dimension 3. By using
958:-atlas. In particular, a
2272:Riemann curvature tensor
1381: > 18 in a
1058:âatlases there exists a
1003:âatlas on a given set a
984:of such atlases are the
36:differentiable structure
2599:Differential structures
1684:Siebenmann, Laurence C.
1092:Hilbert's fifth problem
990:differential structures
2064:Manifold with boundary
1779:Differential structure
1546:Mathematical structure
1530:
1439:
975:differential structure
907:
854:
797:
753:
658:
586:
518:
445:
383:
326:
264:
222:
145:
122:) and open subsets of
94:differential structure
32:differential structure
16:Mathematical structure
1637:Annals of Mathematics
1579:Differential Topology
1531:
1440:
1338:Laurent C. Siebenmann
1076:topological manifolds
908:
855:
798:
754:
659:
587:
519:
446:
384:
327:
273:Consider two charts:
265:
223:
146:
77:For a natural number
63:differential calculus
55:differential manifold
2211:Covariant derivative
1762:Topological manifold
1449:
1390:
870:
817:
766:
669:
606:
529:
461:
399:
337:
280:
245:
158:
126:
106:, which is a set of
65:on the manifold. If
59:topological manifold
2245:Exterior derivative
1847:AtiyahâSinger index
1796:Riemannian manifold
1581:, Springer (1997),
1300:Poincaré conjecture
1088:Donaldson's theorem
1066:â, structures in a
982:equivalence classes
966:-compatible with a
729:
114:between subsets of
96:is defined using a
2551:Secondary calculus
2505:Singularity theory
2460:Parallel transport
2228:De Rham cohomology
1867:Generalized Stokes
1618:10.1007/BF02565940
1526:
1435:
1330:obstruction theory
1047:âatlases whenever
943:-atlas defining a
903:
850:
793:
749:
712:
654:
582:
514:
441:
379:
322:
260:
218:
141:
2586:
2585:
2468:
2467:
2233:Differential form
1887:Whitney embedding
1821:Differential form
1640:. Second Series.
1289:
1288:
2606:
2578:Stratified space
2536:Fréchet manifold
2250:Interior product
2143:
2142:
1840:
1736:
1729:
1722:
1713:
1712:
1706:
1705:
1693:
1680:Kirby, Robion C.
1676:
1670:
1669:
1628:
1622:
1621:
1600:Kervaire, Michel
1596:
1590:
1572:
1535:
1533:
1532:
1527:
1501:
1500:
1488:
1487:
1478:
1477:
1465:
1464:
1459:
1458:
1444:
1442:
1441:
1436:
1422:
1421:
1412:
1411:
1402:
1401:
1383:simply connected
1157:
1156:
1015:For any integer
912:
910:
909:
904:
902:
901:
885:
884:
859:
857:
856:
851:
849:
848:
832:
831:
802:
800:
799:
794:
792:
791:
778:
777:
758:
756:
755:
750:
745:
741:
740:
728:
720:
706:
705:
684:
683:
663:
661:
660:
655:
653:
652:
637:
636:
621:
620:
591:
589:
588:
583:
578:
574:
573:
557:
556:
544:
543:
523:
521:
520:
515:
510:
506:
505:
489:
488:
476:
475:
450:
448:
447:
442:
440:
439:
427:
426:
414:
413:
388:
386:
385:
380:
375:
374:
362:
361:
349:
348:
331:
329:
328:
323:
318:
317:
305:
304:
292:
291:
269:
267:
266:
261:
259:
258:
253:
227:
225:
224:
219:
217:
216:
211:
202:
201:
189:
188:
170:
169:
150:
148:
147:
142:
140:
139:
134:
2614:
2613:
2609:
2608:
2607:
2605:
2604:
2603:
2589:
2588:
2587:
2582:
2521:Banach manifold
2514:Generalizations
2509:
2464:
2401:
2298:
2260:Ricci curvature
2216:Cotangent space
2194:
2132:
1974:
1968:
1927:Exponential map
1891:
1836:
1830:
1750:
1740:
1710:
1709:
1702:
1677:
1673:
1650:10.2307/1969769
1632:Moise, Edwin E.
1629:
1625:
1597:
1593:
1573:
1569:
1564:
1542:
1496:
1492:
1483:
1482:
1473:
1469:
1460:
1454:
1453:
1452:
1450:
1447:
1446:
1417:
1413:
1407:
1406:
1397:
1393:
1391:
1388:
1387:
1380:
1373:
1350:Michel Kervaire
1318:
1132:
1126:
1084:Kervaire (1960)
1078:which admit no
1033:Hassler Whitney
1019:> 0 and any
1013:
962:-atlas that is
894:
890:
877:
873:
871:
868:
867:
841:
837:
824:
820:
818:
815:
814:
787:
783:
773:
769:
767:
764:
763:
730:
721:
716:
711:
707:
701:
697:
676:
672:
670:
667:
666:
645:
641:
629:
625:
613:
609:
607:
604:
603:
566:
562:
558:
552:
548:
536:
532:
530:
527:
526:
498:
494:
490:
484:
480:
468:
464:
462:
459:
458:
435:
431:
422:
418:
406:
402:
400:
397:
396:
370:
366:
357:
353:
344:
340:
338:
335:
334:
313:
309:
300:
296:
287:
283:
281:
278:
277:
254:
249:
248:
246:
243:
242:
212:
207:
206:
197:
193:
184:
180:
165:
161:
159:
156:
155:
135:
130:
129:
127:
124:
123:
75:
17:
12:
11:
5:
2612:
2602:
2601:
2584:
2583:
2581:
2580:
2575:
2570:
2565:
2560:
2559:
2558:
2548:
2543:
2538:
2533:
2528:
2523:
2517:
2515:
2511:
2510:
2508:
2507:
2502:
2497:
2492:
2487:
2482:
2476:
2474:
2470:
2469:
2466:
2465:
2463:
2462:
2457:
2452:
2447:
2442:
2437:
2432:
2427:
2422:
2417:
2411:
2409:
2403:
2402:
2400:
2399:
2394:
2389:
2384:
2379:
2374:
2369:
2359:
2354:
2349:
2339:
2334:
2329:
2324:
2319:
2314:
2308:
2306:
2300:
2299:
2297:
2296:
2291:
2286:
2285:
2284:
2274:
2269:
2268:
2267:
2257:
2252:
2247:
2242:
2241:
2240:
2230:
2225:
2224:
2223:
2213:
2208:
2202:
2200:
2196:
2195:
2193:
2192:
2187:
2182:
2177:
2176:
2175:
2165:
2160:
2155:
2149:
2147:
2140:
2134:
2133:
2131:
2130:
2125:
2115:
2110:
2096:
2091:
2086:
2081:
2076:
2074:Parallelizable
2071:
2066:
2061:
2060:
2059:
2049:
2044:
2039:
2034:
2029:
2024:
2019:
2014:
2009:
2004:
1994:
1984:
1978:
1976:
1970:
1969:
1967:
1966:
1961:
1956:
1954:Lie derivative
1951:
1949:Integral curve
1946:
1941:
1936:
1935:
1934:
1924:
1919:
1918:
1917:
1910:Diffeomorphism
1907:
1901:
1899:
1893:
1892:
1890:
1889:
1884:
1879:
1874:
1869:
1864:
1859:
1854:
1849:
1843:
1841:
1832:
1831:
1829:
1828:
1823:
1818:
1813:
1808:
1803:
1798:
1793:
1788:
1787:
1786:
1781:
1771:
1770:
1769:
1758:
1756:
1755:Basic concepts
1752:
1751:
1739:
1738:
1731:
1724:
1716:
1708:
1707:
1700:
1671:
1623:
1591:
1575:Hirsch, Morris
1566:
1565:
1563:
1560:
1559:
1558:
1553:
1548:
1541:
1538:
1525:
1522:
1519:
1516:
1513:
1510:
1507:
1504:
1499:
1495:
1491:
1486:
1481:
1476:
1472:
1468:
1463:
1457:
1434:
1431:
1428:
1425:
1420:
1416:
1410:
1405:
1400:
1396:
1378:
1371:
1326:Edwin E. Moise
1317:
1314:
1287:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1262:
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1222:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1179:
1176:
1173:
1170:
1167:
1164:
1161:
1152:exotic spheres
1128:Main article:
1125:
1122:
1099:homeomorphisms
1012:
1009:
914:
913:
900:
897:
893:
888:
883:
880:
876:
861:
860:
847:
844:
840:
835:
830:
827:
823:
790:
786:
781:
776:
772:
760:
759:
748:
744:
739:
736:
733:
727:
724:
719:
715:
710:
704:
700:
696:
693:
690:
687:
682:
679:
675:
664:
651:
648:
644:
640:
635:
632:
628:
624:
619:
616:
612:
597:transition map
593:
592:
581:
577:
572:
569:
565:
561:
555:
551:
547:
542:
539:
535:
524:
513:
509:
504:
501:
497:
493:
487:
483:
479:
474:
471:
467:
452:
451:
438:
434:
430:
425:
421:
417:
412:
409:
405:
390:
389:
378:
373:
369:
365:
360:
356:
352:
347:
343:
332:
321:
316:
312:
308:
303:
299:
295:
290:
286:
257:
252:
229:
228:
215:
210:
205:
200:
196:
192:
187:
183:
179:
176:
173:
168:
164:
138:
133:
74:
71:
15:
9:
6:
4:
3:
2:
2611:
2600:
2597:
2596:
2594:
2579:
2576:
2574:
2573:Supermanifold
2571:
2569:
2566:
2564:
2561:
2557:
2554:
2553:
2552:
2549:
2547:
2544:
2542:
2539:
2537:
2534:
2532:
2529:
2527:
2524:
2522:
2519:
2518:
2516:
2512:
2506:
2503:
2501:
2498:
2496:
2493:
2491:
2488:
2486:
2483:
2481:
2478:
2477:
2475:
2471:
2461:
2458:
2456:
2453:
2451:
2448:
2446:
2443:
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2438:
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2423:
2421:
2418:
2416:
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2410:
2408:
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2383:
2380:
2378:
2375:
2373:
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2364:
2360:
2358:
2355:
2353:
2350:
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2340:
2338:
2335:
2333:
2330:
2328:
2325:
2323:
2320:
2318:
2315:
2313:
2310:
2309:
2307:
2305:
2301:
2295:
2294:Wedge product
2292:
2290:
2287:
2283:
2280:
2279:
2278:
2275:
2273:
2270:
2266:
2263:
2262:
2261:
2258:
2256:
2253:
2251:
2248:
2246:
2243:
2239:
2238:Vector-valued
2236:
2235:
2234:
2231:
2229:
2226:
2222:
2219:
2218:
2217:
2214:
2212:
2209:
2207:
2204:
2203:
2201:
2197:
2191:
2188:
2186:
2183:
2181:
2178:
2174:
2171:
2170:
2169:
2168:Tangent space
2166:
2164:
2161:
2159:
2156:
2154:
2151:
2150:
2148:
2144:
2141:
2139:
2135:
2129:
2126:
2124:
2120:
2116:
2114:
2111:
2109:
2105:
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2097:
2095:
2092:
2090:
2087:
2085:
2082:
2080:
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2075:
2072:
2070:
2067:
2065:
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2055:
2054:
2053:
2050:
2048:
2045:
2043:
2040:
2038:
2035:
2033:
2030:
2028:
2025:
2023:
2020:
2018:
2015:
2013:
2010:
2008:
2005:
2003:
1999:
1995:
1993:
1989:
1985:
1983:
1980:
1979:
1977:
1971:
1965:
1962:
1960:
1957:
1955:
1952:
1950:
1947:
1945:
1942:
1940:
1937:
1933:
1932:in Lie theory
1930:
1929:
1928:
1925:
1923:
1920:
1916:
1913:
1912:
1911:
1908:
1906:
1903:
1902:
1900:
1898:
1894:
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1878:
1875:
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1868:
1865:
1863:
1860:
1858:
1855:
1853:
1850:
1848:
1845:
1844:
1842:
1839:
1835:Main results
1833:
1827:
1824:
1822:
1819:
1817:
1816:Tangent space
1814:
1812:
1809:
1807:
1804:
1802:
1799:
1797:
1794:
1792:
1789:
1785:
1782:
1780:
1777:
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1775:
1772:
1768:
1765:
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1757:
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1744:
1737:
1732:
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1725:
1723:
1718:
1717:
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1701:0-691-08190-5
1697:
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1675:
1667:
1663:
1659:
1655:
1651:
1647:
1644:(1): 96â114.
1643:
1639:
1638:
1633:
1627:
1619:
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1611:
1607:
1606:
1601:
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1588:
1587:0-387-90148-5
1584:
1580:
1576:
1571:
1567:
1557:
1556:Exotic sphere
1554:
1552:
1549:
1547:
1544:
1543:
1537:
1523:
1520:
1517:
1514:
1508:
1502:
1497:
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1479:
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1466:
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1423:
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1398:
1394:
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1362:
1358:
1355:
1354:Morris Hirsch
1351:
1347:
1343:
1342:PL structures
1339:
1335:
1331:
1327:
1323:
1313:
1311:
1307:
1306:
1301:
1298:
1294:
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1260:
1257:
1254:
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1245:
1242:
1239:
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1233:
1230:
1227:
1225:Smooth types
1224:
1223:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
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1183:
1180:
1177:
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1168:
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1159:
1158:
1155:
1153:
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1145:
1141:
1137:
1131:
1130:Exotic sphere
1121:
1119:
1118:
1112:
1108:
1104:
1100:
1095:
1093:
1089:
1085:
1081:
1077:
1073:
1069:
1065:
1061:
1057:
1054:
1050:
1046:
1042:
1038:
1034:
1030:
1026:
1023:âdimensional
1022:
1018:
1008:
1006:
1002:
997:
995:
991:
989:
983:
980:
976:
974:
969:
965:
961:
957:
953:
951:
946:
942:
938:
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930:
926:
922:
921:
898:
895:
891:
886:
881:
878:
874:
866:
865:
864:
845:
842:
838:
833:
828:
825:
821:
813:
812:
811:
809:
807:
788:
784:
779:
774:
770:
746:
742:
737:
734:
731:
725:
722:
717:
713:
708:
702:
698:
694:
688:
680:
677:
673:
665:
649:
646:
642:
633:
630:
626:
622:
617:
614:
610:
602:
601:
600:
598:
579:
575:
570:
567:
563:
559:
553:
549:
545:
540:
537:
533:
525:
511:
507:
502:
499:
495:
491:
485:
481:
477:
472:
469:
465:
457:
456:
455:
436:
432:
428:
423:
419:
415:
410:
407:
403:
395:
394:
393:
376:
371:
367:
358:
354:
350:
345:
341:
333:
319:
314:
310:
301:
297:
293:
288:
284:
276:
275:
274:
271:
255:
239:
237:
235:
213:
203:
198:
194:
185:
181:
177:
174:
171:
166:
162:
154:
153:
152:
136:
121:
117:
113:
109:
105:
104:
100:
95:
93:
90:-dimensional
89:
84:
80:
70:
68:
64:
60:
57:, which is a
56:
53:-dimensional
52:
48:
44:
41:
37:
33:
30:
26:
22:
2500:Moving frame
2495:Morse theory
2485:Gauge theory
2277:Tensor field
2206:Closed/Exact
2185:Vector field
2153:Distribution
2094:Hypercomplex
2089:Quaternionic
1826:Vector field
1784:Smooth atlas
1778:
1689:
1674:
1641:
1635:
1626:
1609:
1603:
1594:
1578:
1570:
1375:
1368:
1365:Betti number
1359:
1334:Robion Kirby
1319:
1309:
1303:
1296:
1292:
1290:
1147:
1143:
1139:
1135:
1133:
1116:
1110:
1106:
1102:
1096:
1079:
1071:
1067:
1063:
1059:
1055:
1052:
1048:
1044:
1040:
1036:
1028:
1024:
1020:
1016:
1014:
1004:
1000:
998:
987:
985:
978:
972:
971:
967:
963:
959:
955:
949:
948:
944:
940:
936:
932:
928:
924:
919:
915:
862:
805:
804:
761:
594:
453:
391:
272:
240:
233:
232:
230:
119:
115:
111:
98:
97:
91:
87:
86:
82:
78:
76:
66:
50:
46:
42:
35:
31:
24:
18:
2445:Levi-Civita
2435:Generalized
2407:Connections
2357:Lie algebra
2289:Volume form
2190:Vector flow
2163:Pushforward
2158:Lie bracket
2057:Lie algebra
2022:G-structure
1811:Pushforward
1791:Submanifold
1612:: 257â270.
1361:Dimension 4
1346:John Milnor
952:-equivalent
808:-compatible
762:Two charts
236:-compatible
29:dimensional
21:mathematics
2568:Stratifold
2526:Diffeology
2322:Associated
2123:Symplectic
2108:Riemannian
2037:Hyperbolic
1964:Submersion
1872:HopfâRinow
1806:Submersion
1801:Smooth map
1562:References
1322:Tibor RadĂł
1160:Dimension
231:which are
108:bijections
73:Definition
2450:Principal
2425:Ehresmann
2382:Subbundle
2372:Principal
2347:Fibration
2327:Cotangent
2199:Covectors
2052:Lie group
2032:Hermitian
1975:manifolds
1944:Immersion
1939:Foliation
1877:Noether's
1862:Frobenius
1857:De Rham's
1852:Darboux's
1743:Manifolds
1509:∗
1503:∖
1480:×
1090:(compare
986:distinct
892:φ
875:φ
785:φ
771:φ
723:−
714:φ
699:φ
674:φ
639:→
611:φ
550:φ
482:φ
429:∩
364:→
342:φ
307:→
285:φ
204:⊂
191:→
178:⊃
163:φ
81:and some
2593:Category
2546:Orbifold
2541:K-theory
2531:Diffiety
2255:Pullback
2069:Oriented
2047:Kenmotsu
2027:Hadamard
1973:Types of
1922:Geodesic
1747:Glossary
1686:(1977).
1551:Exotic R
1540:See also
1138:âsphere
1053:distinct
1043:maximal
1041:distinct
994:manifold
49:into an
2490:History
2473:Related
2387:Tangent
2365:)
2345:)
2312:Adjoint
2304:Bundles
2282:density
2180:Torsion
2146:Vectors
2138:Tensors
2121:)
2106:)
2102:,
2100:Pseudoâ
2079:Poisson
2012:Finsler
2007:Fibered
2002:Contact
2000:)
1992:Complex
1990:)
1959:Section
1666:0048805
1658:1969769
1115:exotic
992:of the
110:called
38:) on a
2455:Vector
2440:Koszul
2420:Cartan
2415:Affine
2397:Vector
2392:Tensor
2377:Spinor
2367:Normal
2363:Stable
2317:Affine
2221:bundle
2173:bundle
2119:Almost
2042:KĂ€hler
1998:Almost
1988:Almost
1982:Closed
1882:Sard's
1838:(list)
1698:
1664:
1656:
1585:
1367:
1352:, and
1297:smooth
1282:523264
112:charts
45:makes
2563:Sheaf
2337:Fiber
2113:Rizza
2084:Prime
1915:Local
1905:Curve
1767:Atlas
1654:JSTOR
1302:(see
1270:16256
1105:with
923:. If
916:have
103:atlas
23:, an
2430:Form
2332:Dual
2265:flow
2128:Tame
2104:Subâ
2017:Flat
1897:Maps
1696:ISBN
1583:ISBN
1336:and
803:are
595:The
34:(or
2352:Jet
1646:doi
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1120:.
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2098:(
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1986:(
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1111:n
1107:n
1103:R
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1025:C
1021:n
1017:k
1005:C
1001:C
988:C
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968:C
964:C
960:C
956:C
950:C
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937:C
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925:k
920:k
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