1354:. However, when is this condition sufficient? That is, when can we geometrically cancel handles if this condition is true? The answer lies in carefully analyzing when the manifold remains simply-connected after removing the attaching and belt spheres in question, and finding an embedded disk using the
1927:
1033:
124:
2310:
1095:
1264:
208:, which geometrically untangles homologically-untangled spheres of complementary dimension in a manifold of dimension >4. An informal reason why manifolds of dimension 3 or 4 are unusually hard is that
739:
1764:
1555:
904:
196:
Before Smale proved this theorem, mathematicians became stuck while trying to understand manifolds of dimension 3 or 4, and assumed that the higher-dimensional cases were even harder. The
1850:
814:
2187:
2154:
2066:
1855:
1296:
1189:
1714:
1646:
1329:
1147:
940:
1682:
200:-cobordism theorem showed that (simply connected) manifolds of dimension at least 5 are much easier than those of dimension 3 or 4. The proof of the theorem depends on the "
2484:
1981:
1610:
1442:
594:
945:
562:
1483:
765:
494:
2004:
1954:
1791:
1352:
841:
621:
1983:, which is an integer matrix, restricts to an invertible morphism which can thus be diagonalized via elementary row operations (handle sliding) and must have only
2006:
on the diagonal because it is invertible. Thus, all handles are paired with a single other cancelling handle yielding a decomposition with no handles.
627:-handle. The goal of the proof is to find a handle decomposition with no handles at all so that integrating the non-zero gradient vector field of
353:-cobordant. However, if the intersection form is odd there are non-homeomorphic 4-manifolds with the same intersection form (distinguished by the
85:
2226:
642:
First, we want to rearrange all handles by order so that lower order handles are attached first. The question is thus when can we slide an
1038:
1194:
661:
17:
2428:
Perelman, Grisha (2003-07-17). "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds".
1719:
184:
and is a fundamental result in the theory of high-dimensional manifolds. For a start, it almost immediately proves the
1492:
185:
2560:
2523:
1612:
corresponds to a geometric operation. Indeed, it isn't hard to show (best done by drawing a picture) that sliding a
850:
209:
441:-cobordism theorem is trivially true: the interval is the only connected cobordism between connected 0-manifolds.
2597:
1799:
349:
proved that closed oriented simply-connected topological four-manifolds with equivalent intersection forms are
2581:
1485:
which we can fatten to a cancelling pair as desired, so long as we can embed this disk into the boundary of
773:
2576:
2507:
2448:
Note that identifying the
Whitehead groups of the various manifolds requires that one choose base points
2157:
2083:
1922:{\displaystyle C_{k}\cong \operatorname {coker} \partial _{k+1}\oplus \operatorname {im} \partial _{k+1}}
2386:
Perelman, Grisha (2002-11-11). "The entropy formula for the Ricci flow and its geometric applications".
2166:
2133:
2045:
430:-cobordism theorem is vacuously true, since there is no closed simply-connected 1-dimensional manifold.
2541:
354:
1269:
1156:
1687:
1619:
1355:
1301:
1120:
913:
404:
260:
201:
163:
1655:
2451:
1959:
1588:
1405:
1028:{\displaystyle \sum _{\beta }\langle h_{\alpha }^{k}\mid h_{\beta }^{k-1}\rangle h_{\beta }^{k-1}}
2602:
2571:
511:
396:
377:
1455:
744:
455:
362:
35:
1362:
must be at least 5. Moreover, during the proof one requires that the cobordism has no 0-,1-,
567:
1986:
1932:
1769:
1334:
819:
599:
497:
266:
169:
76:
2518:. Princeton Mathematical Series. Vol. 39. Princeton, NJ: Princeton University Press.
376:-cobordism theorem for smooth manifolds has not been proved and, due to the 3-dimensional
8:
2566:
S. Smale, "On the structure of manifolds" Amer. J. Math., 84 (1962) pp. 387â399
412:
365:
with the same homotopy type are not homeomorphic but both have intersection form of (1).
2512:
2429:
2408:
2387:
31:
2556:
2519:
2220:
2115:
2031:
1565:
is either 0 or 1. Finally, by considering the negative of the given Morse function, â
450:
400:
2503:
2320:
408:
381:
380:, is equivalent to the hard open question of whether the 4-sphere has non-standard
283:
2552:
254:
205:
157:
2091:
2362:
1796:
The proof of the theorem now follows: the handle chain complex is exact since
2591:
2544:, Princeton, NJ, 1965. v+116 pp. This gives the proof for smooth manifolds.
2407:
Perelman, Grisha (2003-03-10). "Ricci flow with surgery on three-manifolds".
2095:
346:
177:
119:{\displaystyle M\hookrightarrow W\quad {\mbox{and}}\quad N\hookrightarrow W}
2207:-cobordism theorem is that the isomorphism classes of this groupoid (up to
181:
2533:
2087:
2305:{\displaystyle \pi \cong \pi _{1}(M)\cong \pi _{1}(W)\cong \pi _{1}(N).}
1569:, we can turn the handle decomposition upside down and also remove the
416:
1585:
Finally, we want to make sure that doing row and column operations on
2434:
2413:
2392:
2189:
is also a simple homotopy equivalenceâthat follows from the theorem.
43:
1090:{\displaystyle \langle h_{\alpha }^{k}\mid h_{\beta }^{k-1}\rangle }
2197:
1259:{\displaystyle \partial _{k+1}h_{\beta }^{k+1}=\pm h_{\alpha }^{k}}
54:
2030:-cobordisms need not be cylinders; the obstruction is exactly the
1370: + 1)-handles which is obtained by the next technique.
1113:
Next, we want to "cancel" handles. The idea is that attaching a
2216:
411:
in a series of three papers in 2002 and 2003, where he follows
215:
1378:
The idea of handle trading is to create a cancelling pair of (
1398: + 2)-handle. To do this, consider the core of the
650:-handle? This can be done by a radial isotopy so long as the
631:
gives the desired diffeomorphism to the trivial cobordism.
1149:
might create a hole that can be filled in by attaching a (
212:
in lower dimensions, which have no room for entanglement.
2343:
Wall, C.T.C. (1964). "On simply-connected 4-manifolds".
298:Ă . The isomorphism can be chosen to be the identity on
2163:
Note that one need not assume that the other inclusion
100:
2454:
2229:
2169:
2136:
2048:
1989:
1962:
1935:
1858:
1802:
1772:
1722:
1690:
1658:
1622:
1591:
1495:
1458:
1408:
1337:
1304:
1272:
1197:
1159:
1123:
1041:
948:
916:
853:
822:
776:
747:
734:{\displaystyle (i-1)+(n-j)\leq \dim \partial W-1=n-1}
664:
602:
570:
514:
504:, i.e., if there is a single critical point of index
458:
88:
2530:(This does the theorem for topological 4-manifolds.)
345:-cobordism theorem is false. This can be seen since
2511:
2478:
2363:"Millennium Problems | Clay Mathematics Institute"
2304:
2181:
2148:
2130:The torsion vanishes if and only if the inclusion
2060:
1998:
1975:
1948:
1921:
1844:
1785:
1758:
1708:
1676:
1640:
1604:
1549:
1477:
1436:
1346:
1323:
1290:
1258:
1183:
1141:
1089:
1027:
934:
898:
835:
808:
759:
733:
615:
588:
556:
488:
118:
1759:{\displaystyle h_{\alpha }^{k}\pm h_{\beta }^{k}}
634:This is achieved through a series of techniques.
305:This means that the homotopy equivalence between
2589:
2547:Rourke, Colin Patrick; Sanderson, Brian Joseph,
1550:{\displaystyle \dim \partial W-1=n-1\geq 2(k+1)}
1358:. This analysis leads to the requirement that
899:{\displaystyle \partial _{k}:C_{k}\to C_{k-1}}
2502:
332:
1084:
1042:
1001:
959:
2563:. This proves the theorem for PL manifolds.
1394: + 1)-handle leaving behind the (
658:belt sphere do not intersect. We thus want
2345:Journal of the London Mathematical Society
2156:is not just a homotopy equivalence, but a
2009:
2549:Introduction to piecewise-linear topology
2433:
2412:
2391:
2098:, states (assumptions as above but where
1845:{\displaystyle H_{*}(W,M;\mathbb {Z} )=0}
1829:
2540:, notes by L. Siebenmann and J. Sondow,
2427:
2406:
2385:
2114:-cobordism is a cylinder if and only if
1386: + 2)-handles so that a given
770:We then define the handle chain complex
395:-cobordism theorem is equivalent to the
14:
2590:
2569:
143:-cobordism to be trivial, i.e., to be
809:{\displaystyle (C_{*},\partial _{*})}
2342:
1577: + 1)-handles as desired.
2538:Lectures on the h-cobordism theorem
155:refers to any of the categories of
139:gives sufficient conditions for an
24:
2182:{\displaystyle N\hookrightarrow W}
2149:{\displaystyle M\hookrightarrow W}
2061:{\displaystyle M\hookrightarrow W}
1964:
1904:
1879:
1593:
1502:
1452:-cobordism. Thus, there is a disk
1306:
1199:
1097:is the intersection number of the
855:
794:
707:
25:
2614:
2026:are simply connected is dropped,
843:be the free abelian group on the
444:
1291:{\displaystyle (\alpha ,\beta )}
1184:{\displaystyle h_{\beta }^{k+1}}
176:The theorem was first proved by
2106:need not be simply connected):
1709:{\displaystyle h_{\alpha }^{k}}
1641:{\displaystyle h_{\alpha }^{k}}
1402:-handle which is an element in
1324:{\displaystyle \partial _{k+1}}
1142:{\displaystyle h_{\alpha }^{k}}
935:{\displaystyle h_{\alpha }^{k}}
564:, then the ascending cobordism
186:generalized Poincaré conjecture
106:
98:
2442:
2421:
2400:
2379:
2355:
2336:
2296:
2290:
2274:
2268:
2252:
2246:
2203:Then a finer statement of the
2173:
2140:
2052:
1833:
1813:
1677:{\displaystyle h_{\beta }^{k}}
1561:is at least 5 this means that
1544:
1532:
1444:. This group is trivial since
1431:
1419:
1285:
1273:
877:
803:
777:
695:
683:
677:
665:
551:
548:
531:
528:
483:
471:
468:
110:
92:
13:
1:
2496:
2479:{\displaystyle m\in M,n\in N}
1976:{\displaystyle \partial _{k}}
1605:{\displaystyle \partial _{k}}
1437:{\displaystyle \pi _{k}(W,M)}
1105: â 1)-belt sphere.
191:
236: + 1)-dimensional
147:-isomorphic to the cylinder
42: + 1)-dimensional
7:
2577:Encyclopedia of Mathematics
2551:, Springer Study Edition,
2314:
2158:simple homotopy equivalence
2086:), proved independently by
2084:simple-homotopy equivalence
1489:. This embedding exists if
1101:-attaching sphere and the (
129:are homotopy equivalences.
10:
2619:
2555:, Berlin-New York, 1982.
2542:Princeton University Press
1390:-handle cancels with the (
333:Lower dimensional versions
180:for which he received the
654:attaching sphere and the
216:Precise statement of the
2330:
1557:. Since we are assuming
1191:. This would imply that
557:{\displaystyle f^{-1}()}
79:) if the inclusion maps
2570:Rudyak, Yu.B. (2001) ,
2514:Topology of 4-manifolds
2018:If the assumption that
1478:{\displaystyle D^{k+1}}
1298:entry in the matrix of
1153: + 1)-handle
760:{\displaystyle i\leq j}
741:which is equivalent to
638:1) Handle rearrangement
489:{\displaystyle f:W\to }
210:the trick fails to work
2480:
2306:
2183:
2150:
2062:
2000:
1977:
1950:
1923:
1846:
1787:
1760:
1710:
1678:
1642:
1606:
1551:
1479:
1438:
1382: + 1)- and (
1348:
1325:
1292:
1260:
1185:
1143:
1109:2) Handle cancellation
1091:
1029:
936:
900:
847:-handles and defining
837:
810:
761:
735:
617:
590:
589:{\displaystyle W_{c'}}
558:
490:
325:Ă ) is homotopic to a
228:be at least 5 and let
120:
2598:Differential topology
2481:
2307:
2184:
2151:
2063:
2001:
1999:{\displaystyle \pm 1}
1978:
1951:
1949:{\displaystyle C_{k}}
1924:
1847:
1788:
1786:{\displaystyle C_{k}}
1761:
1711:
1679:
1643:
1607:
1552:
1480:
1439:
1349:
1347:{\displaystyle \pm 1}
1326:
1293:
1261:
1186:
1144:
1092:
1030:
937:
901:
838:
836:{\displaystyle C_{k}}
811:
762:
736:
618:
616:{\displaystyle W_{c}}
591:
559:
491:
363:fake projective plane
357:class). For example,
121:
36:differential topology
2452:
2227:
2167:
2134:
2046:
1987:
1960:
1933:
1856:
1800:
1770:
1720:
1688:
1656:
1620:
1589:
1493:
1456:
1406:
1335:
1302:
1270:
1195:
1157:
1121:
1039:
946:
914:
851:
820:
774:
745:
662:
600:
568:
512:
498:handle decomposition
456:
407:) and was proved by
403:in 1904 (one of the
86:
77:homotopy equivalence
2504:Freedman, Michael H
2219:for the respective
2196:-cobordisms form a
2042:) of the inclusion
1755:
1737:
1705:
1673:
1637:
1255:
1234:
1180:
1138:
1083:
1059:
1024:
1000:
976:
931:
413:Richard S. Hamilton
405:Millennium Problems
397:Poincaré conjecture
378:Poincaré conjecture
240:-cobordism between
27:Concept in topology
18:S-cobordism theorem
2476:
2302:
2179:
2146:
2076:-cobordism theorem
2058:
2014:-cobordism theorem
1996:
1973:
1946:
1919:
1842:
1783:
1756:
1741:
1723:
1706:
1691:
1674:
1659:
1638:
1623:
1602:
1547:
1475:
1434:
1344:
1321:
1288:
1256:
1241:
1214:
1181:
1160:
1139:
1124:
1087:
1063:
1045:
1025:
1004:
980:
962:
958:
932:
917:
896:
833:
806:
757:
731:
613:
586:
554:
486:
220:-cobordism theorem
137:-cobordism theorem
116:
104:
32:geometric topology
2215:-cobordisms) are
2116:Whitehead torsion
2032:Whitehead torsion
1766:in the basis for
1581:4) Handle sliding
1374:3) Handle trading
949:
646:-handle off of a
596:is obtained from
415:'s program using
382:smooth structures
103:
16:(Redirected from
2610:
2584:
2529:
2517:
2491:
2490:connecting them.
2485:
2483:
2482:
2477:
2446:
2440:
2439:
2437:
2425:
2419:
2418:
2416:
2404:
2398:
2397:
2395:
2383:
2377:
2376:
2374:
2373:
2367:www.claymath.org
2359:
2353:
2352:
2340:
2311:
2309:
2308:
2303:
2289:
2288:
2267:
2266:
2245:
2244:
2221:Whitehead groups
2211:-isomorphism of
2188:
2186:
2185:
2180:
2155:
2153:
2152:
2147:
2067:
2065:
2064:
2059:
2005:
2003:
2002:
1997:
1982:
1980:
1979:
1974:
1972:
1971:
1955:
1953:
1952:
1947:
1945:
1944:
1928:
1926:
1925:
1920:
1918:
1917:
1893:
1892:
1868:
1867:
1851:
1849:
1848:
1843:
1832:
1812:
1811:
1792:
1790:
1789:
1784:
1782:
1781:
1765:
1763:
1762:
1757:
1754:
1749:
1736:
1731:
1715:
1713:
1712:
1707:
1704:
1699:
1683:
1681:
1680:
1675:
1672:
1667:
1647:
1645:
1644:
1639:
1636:
1631:
1611:
1609:
1608:
1603:
1601:
1600:
1556:
1554:
1553:
1548:
1484:
1482:
1481:
1476:
1474:
1473:
1443:
1441:
1440:
1435:
1418:
1417:
1353:
1351:
1350:
1345:
1330:
1328:
1327:
1322:
1320:
1319:
1297:
1295:
1294:
1289:
1265:
1263:
1262:
1257:
1254:
1249:
1233:
1222:
1213:
1212:
1190:
1188:
1187:
1182:
1179:
1168:
1148:
1146:
1145:
1140:
1137:
1132:
1096:
1094:
1093:
1088:
1082:
1071:
1058:
1053:
1034:
1032:
1031:
1026:
1023:
1012:
999:
988:
975:
970:
957:
941:
939:
938:
933:
930:
925:
905:
903:
902:
897:
895:
894:
876:
875:
863:
862:
842:
840:
839:
834:
832:
831:
815:
813:
812:
807:
802:
801:
789:
788:
766:
764:
763:
758:
740:
738:
737:
732:
622:
620:
619:
614:
612:
611:
595:
593:
592:
587:
585:
584:
583:
563:
561:
560:
555:
547:
527:
526:
495:
493:
492:
487:
409:Grigori Perelman
355:Kirby-Siebenmann
284:simply connected
248:in the category
164:piecewise linear
125:
123:
122:
117:
105:
101:
21:
2618:
2617:
2613:
2612:
2611:
2609:
2608:
2607:
2588:
2587:
2553:Springer-Verlag
2526:
2499:
2494:
2453:
2450:
2449:
2447:
2443:
2426:
2422:
2405:
2401:
2384:
2380:
2371:
2369:
2361:
2360:
2356:
2341:
2337:
2333:
2317:
2284:
2280:
2262:
2258:
2240:
2236:
2228:
2225:
2224:
2192:Categorically,
2168:
2165:
2164:
2135:
2132:
2131:
2071:Precisely, the
2047:
2044:
2043:
2016:
1988:
1985:
1984:
1967:
1963:
1961:
1958:
1957:
1956:are free. Then
1940:
1936:
1934:
1931:
1930:
1907:
1903:
1882:
1878:
1863:
1859:
1857:
1854:
1853:
1828:
1807:
1803:
1801:
1798:
1797:
1777:
1773:
1771:
1768:
1767:
1750:
1745:
1732:
1727:
1721:
1718:
1717:
1700:
1695:
1689:
1686:
1685:
1668:
1663:
1657:
1654:
1653:
1632:
1627:
1621:
1618:
1617:
1596:
1592:
1590:
1587:
1586:
1494:
1491:
1490:
1463:
1459:
1457:
1454:
1453:
1413:
1409:
1407:
1404:
1403:
1336:
1333:
1332:
1309:
1305:
1303:
1300:
1299:
1271:
1268:
1267:
1250:
1245:
1223:
1218:
1202:
1198:
1196:
1193:
1192:
1169:
1164:
1158:
1155:
1154:
1133:
1128:
1122:
1119:
1118:
1072:
1067:
1054:
1049:
1040:
1037:
1036:
1013:
1008:
989:
984:
971:
966:
953:
947:
944:
943:
926:
921:
915:
912:
911:
884:
880:
871:
867:
858:
854:
852:
849:
848:
827:
823:
821:
818:
817:
797:
793:
784:
780:
775:
772:
771:
746:
743:
742:
663:
660:
659:
623:by attaching a
607:
603:
601:
598:
597:
576:
575:
571:
569:
566:
565:
540:
519:
515:
513:
510:
509:
457:
454:
453:
447:
335:
294:-isomorphic to
222:
206:Hassler Whitney
194:
99:
87:
84:
83:
28:
23:
22:
15:
12:
11:
5:
2616:
2606:
2605:
2603:Surgery theory
2600:
2586:
2585:
2567:
2564:
2545:
2531:
2524:
2498:
2495:
2493:
2492:
2486:and a path in
2475:
2472:
2469:
2466:
2463:
2460:
2457:
2441:
2420:
2399:
2378:
2354:
2334:
2332:
2329:
2328:
2327:
2316:
2313:
2301:
2298:
2295:
2292:
2287:
2283:
2279:
2276:
2273:
2270:
2265:
2261:
2257:
2254:
2251:
2248:
2243:
2239:
2235:
2232:
2178:
2175:
2172:
2145:
2142:
2139:
2128:
2127:
2092:John Stallings
2057:
2054:
2051:
2015:
2008:
1995:
1992:
1970:
1966:
1943:
1939:
1916:
1913:
1910:
1906:
1902:
1899:
1896:
1891:
1888:
1885:
1881:
1877:
1874:
1871:
1866:
1862:
1841:
1838:
1835:
1831:
1827:
1824:
1821:
1818:
1815:
1810:
1806:
1780:
1776:
1753:
1748:
1744:
1740:
1735:
1730:
1726:
1703:
1698:
1694:
1671:
1666:
1662:
1635:
1630:
1626:
1599:
1595:
1546:
1543:
1540:
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1472:
1469:
1466:
1462:
1433:
1430:
1427:
1424:
1421:
1416:
1412:
1343:
1340:
1318:
1315:
1312:
1308:
1287:
1284:
1281:
1278:
1275:
1253:
1248:
1244:
1240:
1237:
1232:
1229:
1226:
1221:
1217:
1211:
1208:
1205:
1201:
1178:
1175:
1172:
1167:
1163:
1136:
1131:
1127:
1086:
1081:
1078:
1075:
1070:
1066:
1062:
1057:
1052:
1048:
1044:
1022:
1019:
1016:
1011:
1007:
1003:
998:
995:
992:
987:
983:
979:
974:
969:
965:
961:
956:
952:
929:
924:
920:
893:
890:
887:
883:
879:
874:
870:
866:
861:
857:
830:
826:
805:
800:
796:
792:
787:
783:
779:
756:
753:
750:
730:
727:
724:
721:
718:
715:
712:
709:
706:
703:
700:
697:
694:
691:
688:
685:
682:
679:
676:
673:
670:
667:
610:
606:
582:
579:
574:
553:
550:
546:
543:
539:
536:
533:
530:
525:
522:
518:
485:
482:
479:
476:
473:
470:
467:
464:
461:
451:Morse function
446:
445:A proof sketch
443:
334:
331:
329:-isomorphism.
232:be a compact (
221:
214:
193:
190:
127:
126:
115:
112:
109:
97:
94:
91:
26:
9:
6:
4:
3:
2:
2615:
2604:
2601:
2599:
2596:
2595:
2593:
2583:
2579:
2578:
2573:
2572:"h-cobordism"
2568:
2565:
2562:
2561:3-540-11102-6
2558:
2554:
2550:
2546:
2543:
2539:
2535:
2532:
2527:
2525:0-691-08577-3
2521:
2516:
2515:
2509:
2505:
2501:
2500:
2489:
2473:
2470:
2467:
2464:
2461:
2458:
2455:
2445:
2436:
2431:
2424:
2415:
2410:
2403:
2394:
2389:
2382:
2368:
2364:
2358:
2350:
2346:
2339:
2335:
2326:
2324:
2319:
2318:
2312:
2299:
2293:
2285:
2281:
2277:
2271:
2263:
2259:
2255:
2249:
2241:
2237:
2233:
2230:
2223:Wh(Ï), where
2222:
2218:
2214:
2210:
2206:
2201:
2199:
2195:
2190:
2176:
2170:
2161:
2159:
2143:
2137:
2125:
2121:
2117:
2113:
2109:
2108:
2107:
2105:
2101:
2097:
2096:Dennis Barden
2093:
2089:
2085:
2081:
2077:
2075:
2069:
2055:
2049:
2041:
2037:
2033:
2029:
2025:
2021:
2013:
2007:
1993:
1990:
1968:
1941:
1937:
1914:
1911:
1908:
1900:
1897:
1894:
1889:
1886:
1883:
1875:
1872:
1869:
1864:
1860:
1839:
1836:
1825:
1822:
1819:
1816:
1808:
1804:
1794:
1778:
1774:
1751:
1746:
1742:
1738:
1733:
1728:
1724:
1701:
1696:
1692:
1669:
1664:
1660:
1651:
1648:over another
1633:
1628:
1624:
1615:
1597:
1583:
1582:
1578:
1576:
1572:
1568:
1564:
1560:
1541:
1538:
1535:
1529:
1526:
1523:
1520:
1517:
1514:
1511:
1508:
1505:
1499:
1496:
1488:
1470:
1467:
1464:
1460:
1451:
1447:
1428:
1425:
1422:
1414:
1410:
1401:
1397:
1393:
1389:
1385:
1381:
1376:
1375:
1371:
1369:
1365:
1361:
1357:
1356:Whitney trick
1341:
1338:
1316:
1313:
1310:
1282:
1279:
1276:
1251:
1246:
1242:
1238:
1235:
1230:
1227:
1224:
1219:
1215:
1209:
1206:
1203:
1176:
1173:
1170:
1165:
1161:
1152:
1134:
1129:
1125:
1116:
1111:
1110:
1106:
1104:
1100:
1079:
1076:
1073:
1068:
1064:
1060:
1055:
1050:
1046:
1020:
1017:
1014:
1009:
1005:
996:
993:
990:
985:
981:
977:
972:
967:
963:
954:
950:
927:
922:
918:
909:
906:by sending a
891:
888:
885:
881:
872:
868:
864:
859:
846:
828:
824:
798:
790:
785:
781:
768:
754:
751:
748:
728:
725:
722:
719:
716:
713:
710:
704:
701:
698:
692:
689:
686:
680:
674:
671:
668:
657:
653:
649:
645:
640:
639:
635:
632:
630:
626:
608:
604:
580:
577:
572:
544:
541:
537:
534:
523:
520:
516:
507:
503:
499:
480:
477:
474:
465:
462:
459:
452:
442:
440:
436:
431:
429:
425:
420:
418:
414:
410:
406:
402:
398:
394:
390:
385:
383:
379:
375:
371:
366:
364:
360:
356:
352:
348:
344:
340:
330:
328:
324:
320:
316:
313:(or, between
312:
308:
303:
301:
297:
293:
289:
285:
281:
277:
273:
269:
268:
263:
262:
257:
256:
251:
247:
243:
239:
235:
231:
227:
219:
213:
211:
207:
203:
202:Whitney trick
199:
189:
187:
183:
179:
178:Stephen Smale
174:
172:
171:
166:
165:
160:
159:
154:
150:
146:
142:
138:
136:
130:
113:
107:
95:
89:
82:
81:
80:
78:
74:
70:
68:
63:
59:
56:
53:-dimensional
52:
48:
45:
41:
37:
33:
19:
2575:
2548:
2537:
2534:Milnor, John
2513:
2508:Quinn, Frank
2487:
2444:
2435:math/0307245
2423:
2414:math/0303109
2402:
2393:math/0211159
2381:
2370:. Retrieved
2366:
2357:
2348:
2344:
2338:
2322:
2212:
2208:
2204:
2202:
2193:
2191:
2162:
2129:
2123:
2119:
2111:
2103:
2099:
2079:
2073:
2072:
2070:
2039:
2035:
2027:
2023:
2019:
2017:
2011:
1795:
1649:
1613:
1584:
1580:
1579:
1574:
1570:
1566:
1562:
1558:
1486:
1449:
1445:
1399:
1395:
1391:
1387:
1383:
1379:
1377:
1373:
1372:
1367:
1363:
1359:
1150:
1114:
1112:
1108:
1107:
1102:
1098:
907:
844:
769:
655:
651:
647:
643:
641:
637:
636:
633:
628:
624:
505:
501:
448:
438:
434:
432:
427:
423:
421:
392:
388:
386:
373:
369:
367:
358:
350:
342:
338:
336:
326:
322:
318:
314:
310:
306:
304:
299:
295:
291:
287:
279:
275:
271:
265:
259:
253:
249:
245:
241:
237:
233:
229:
225:
223:
217:
197:
195:
182:Fields Medal
175:
168:
162:
156:
152:
148:
144:
140:
134:
133:
131:
128:
72:
66:
65:
61:
57:
50:
46:
39:
29:
2126:) vanishes.
2088:Barry Mazur
2082:stands for
1266:and so the
816:by letting
173:manifolds.
170:topological
75:stands for
2592:Categories
2497:References
2372:2016-03-30
2325:-cobordism
1929:since the
496:induces a
417:Ricci flow
399:stated by
270:such that
192:Background
69:-cobordism
2582:EMS Press
2471:∈
2459:∈
2351:: 141â49.
2282:π
2278:≅
2260:π
2256:≅
2238:π
2234:≅
2231:π
2174:↪
2141:↪
2053:↪
1991:±
1965:∂
1905:∂
1901:
1895:⊕
1880:∂
1876:
1870:≅
1809:∗
1747:β
1739:±
1729:α
1697:α
1684:replaces
1665:β
1629:α
1594:∂
1527:≥
1521:−
1509:−
1503:∂
1500:
1411:π
1339:±
1331:would be
1307:∂
1283:β
1277:α
1247:α
1239:±
1220:β
1200:∂
1166:β
1130:α
1085:⟩
1077:−
1069:β
1061:∣
1051:α
1043:⟨
1018:−
1010:β
1002:⟩
994:−
986:β
978:∣
968:α
960:⟨
955:β
951:∑
923:α
889:−
878:→
856:∂
799:∗
795:∂
786:∗
752:≤
726:−
714:−
708:∂
705:
699:≤
690:−
672:−
521:−
469:→
437:= 0, the
426:= 1, the
391:= 2, the
372:= 3, the
341:= 4, the
151:Ă . Here
111:↪
93:↪
55:manifolds
44:cobordism
2510:(1990).
2315:See also
2198:groupoid
1652:-handle
1616:-handle
1117:-handle
1035:, where
910:-handle
581:′
545:′
401:Poincaré
49:between
2217:torsors
1852:. Thus
1573:- and (
1366:-, or (
302:Ă {0}.
286:. Then
2559:
2522:
2094:, and
1448:is an
361:and a
158:smooth
64:is an
38:, an (
2430:arXiv
2409:arXiv
2388:arXiv
2331:Notes
2321:Semi-
2078:(the
1873:coker
264:, or
204:" of
167:, or
71:(the
2557:ISBN
2520:ISBN
2102:and
2022:and
2010:The
433:For
422:For
387:For
368:For
347:Wall
337:For
321:and
317:Ă ,
309:and
282:are
278:and
255:Diff
244:and
224:Let
132:The
60:and
34:and
2118:Ï (
2110:An
2034:Ï (
1716:by
1497:dim
942:to
702:dim
508:in
500:of
290:is
267:Top
102:and
30:In
2594::
2580:,
2574:,
2536:,
2506:;
2365:.
2349:39
2347:.
2200:.
2160:.
2122:,
2090:,
2068:.
2038:,
1898:im
1793:.
767:.
449:A
419:.
384:.
359:CP
274:,
261:PL
258:,
188:.
161:,
2528:.
2488:W
2474:N
2468:n
2465:,
2462:M
2456:m
2438:.
2432::
2417:.
2411::
2396:.
2390::
2375:.
2323:s
2300:.
2297:)
2294:N
2291:(
2286:1
2275:)
2272:W
2269:(
2264:1
2253:)
2250:M
2247:(
2242:1
2213:h
2209:C
2205:s
2194:h
2177:W
2171:N
2144:W
2138:M
2124:M
2120:W
2112:h
2104:N
2100:M
2080:s
2074:s
2056:W
2050:M
2040:M
2036:W
2028:h
2024:N
2020:M
2012:s
1994:1
1969:k
1942:k
1938:C
1915:1
1912:+
1909:k
1890:1
1887:+
1884:k
1865:k
1861:C
1840:0
1837:=
1834:)
1830:Z
1826:;
1823:M
1820:,
1817:W
1814:(
1805:H
1779:k
1775:C
1752:k
1743:h
1734:k
1725:h
1702:k
1693:h
1670:k
1661:h
1650:k
1634:k
1625:h
1614:k
1598:k
1575:n
1571:n
1567:f
1563:k
1559:n
1545:)
1542:1
1539:+
1536:k
1533:(
1530:2
1524:1
1518:n
1515:=
1512:1
1506:W
1487:W
1471:1
1468:+
1465:k
1461:D
1450:h
1446:W
1432:)
1429:M
1426:,
1423:W
1420:(
1415:k
1400:k
1396:k
1392:k
1388:k
1384:k
1380:k
1368:n
1364:n
1360:n
1342:1
1317:1
1314:+
1311:k
1286:)
1280:,
1274:(
1252:k
1243:h
1236:=
1231:1
1228:+
1225:k
1216:h
1210:1
1207:+
1204:k
1177:1
1174:+
1171:k
1162:h
1151:k
1135:k
1126:h
1115:k
1103:k
1099:k
1080:1
1074:k
1065:h
1056:k
1047:h
1021:1
1015:k
1006:h
997:1
991:k
982:h
973:k
964:h
928:k
919:h
908:k
892:1
886:k
882:C
873:k
869:C
865::
860:k
845:k
829:k
825:C
804:)
791:,
782:C
778:(
755:j
749:i
729:1
723:n
720:=
717:1
711:W
696:)
693:j
687:n
684:(
681:+
678:)
675:1
669:i
666:(
656:j
652:i
648:j
644:i
629:f
625:k
609:c
605:W
578:c
573:W
552:)
549:]
542:c
538:,
535:c
532:[
529:(
524:1
517:f
506:k
502:W
484:]
481:b
478:,
475:a
472:[
466:W
463::
460:f
439:h
435:n
428:h
424:n
393:h
389:n
374:h
370:n
351:h
343:h
339:n
327:C
323:N
319:W
315:M
311:N
307:M
300:M
296:M
292:C
288:W
280:N
276:M
272:W
252:=
250:C
246:N
242:M
238:h
234:n
230:W
226:n
218:h
198:h
153:C
149:M
145:C
141:h
135:h
114:W
108:N
96:W
90:M
73:h
67:h
62:N
58:M
51:n
47:W
40:n
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.