702:
919:
2052:
1424:
321:
971:. For each such point, in the limit one obtains a unique singular ASD connection, which becomes a well-defined smooth ASD connection at that point using Uhlenbeck's removable singularity theorem.
1606:
1553:
830:
1725:
646:
172:
1650:
510:
134:
1885:
1780:
1035:
2097:
1280:
1198:
1084:
858:
996:
799:
775:
696:
1932:
709:
This moduli space is non-compact and generically smooth, with singularities occurring only at the points corresponding to reducible connections, of which there are exactly
850:
614:
572:
366:
743:
466:
406:
1905:
969:
1156:
1853:
1679:
672:
2075:
1925:
1824:
1800:
1745:
1487:
1467:
1447:
1238:
1218:
1124:
1104:
1055:
943:
530:
430:
213:
193:
1287:
1750:
It is thus possible to compactify the moduli space as follows: First, cut off each cone at a reducible singularity and glue in a copy of
2266:
Donaldson, S. K. (1983). An application of gauge theory to four-dimensional topology. Journal of
Differential Geometry, 18(2), 279-315.
2275:
Taubes, C. H. (1982). Self-dual Yang–Mills connections on non-self-dual 4-manifolds. Journal of
Differential Geometry, 17(1), 139-170.
225:
137:
2296:
Uhlenbeck, K. K. (1982). Removable singularities in Yang–Mills fields. Communications in
Mathematical Physics, 83(1), 11-29.
1558:
2284:
Uhlenbeck, K. K. (1982). Connections with L p bounds on curvature. Communications in
Mathematical Physics, 83(1), 31-42.
2089:
1492:
2359:
925:
The work of Taubes and
Uhlenbeck essentially concerns constructing sequences of ASD connections on the four-manifold
2312:
804:
216:
2411:
1684:
39:
1681:. An elementary argument that applies to any negative definite quadratic form over the integers tells us that
2416:
2395:
2123:
619:
145:
1611:
471:
532:
is simply-connected with definite intersection form, possibly after changing orientation, one always has
914:{\displaystyle {\mathcal {M}}_{\varepsilon }{\xrightarrow {\quad \cong \quad }}X\times (0,\varepsilon )}
108:
1858:
1753:
1008:
777:
is non-compact, its structure at infinity can be readily described. Namely, there is an open subset of
28:
2047:{\displaystyle {\text{rank}}(Q)=b_{2}(X)=\sigma (X)=\sigma (\bigsqcup n(Q)\mathbb {CP} ^{2})\leq n(Q)}
1243:
1161:
1060:
977:
780:
756:
677:
2421:
2145:
2108:
1003:
835:
36:
577:
535:
329:
712:
435:
375:
2119:
1890:
999:
59:
24:
948:
2335:
2150:
2104:
1129:
2343:
1829:
1655:
62:. If the intersection form is positive (negative) definite, it can be diagonalized to the
8:
651:
2310:
Donaldson, S. K. (1983), "An application of gauge theory to four-dimensional topology",
2103:
1) Any indefinite non-diagonalizable intersection form gives rise to a four-dimensional
1555:, we get that reducible connections modulo gauge are in a 1-1 correspondence with pairs
2242:
2135:
2060:
1910:
1809:
1785:
1730:
1472:
1452:
1432:
1223:
1203:
1158:. Then, reducible connections modulo gauge are in a 1-1 correspondence with splittings
1109:
1089:
1040:
928:
515:
415:
198:
178:
2355:
2246:
2234:
2193:
75:
2339:
2321:
2224:
2183:
2140:
2085:
141:
71:
701:
2369:
2331:
998:
corresponding to reducible connections could also be described: they looked like
750:
746:
87:
63:
51:
2405:
2326:
2238:
2229:
2197:
2188:
2093:
47:
43:
16:
On when a definite intersection form of a smooth 4-manifold is diagonalizable
2212:
2171:
2115:
409:
103:
91:
945:
with curvature becoming infinitely concentrated at any given single point
369:
20:
1419:{\displaystyle 1=k=c_{2}(E)=c_{2}(L\oplus L^{-1})=-Q(c_{1}(L),c_{1}(L))}
2213:"The orientation of Yang-Mills moduli spaces and 4-manifold topology"
1803:
55:
1037:. Furthermore, we can count the number of such singular points. Let
881:
2100:
and
Donaldson's theorem, several interesting results can be seen:
70:. The original version of the theorem required the manifold to be
705:
Cobordism given by Yang–Mills moduli space in
Donaldson's theorem
974:
Donaldson observed that the singular points in the interior of
2092:
is realized as the intersection form of some closed, oriented
316:{\displaystyle \dim {\mathcal {M}}=8k-3(1-b_{1}(X)+b_{+}(X)),}
2172:"An application of gauge theory to four-dimensional topology"
74:, but it was later improved to apply to 4-manifolds with any
1907:
of a four-manifold is a cobordism invariant. Thus, because
2118:, if and only if, their intersection forms have the same
832:, such that for sufficiently small choices of parameter
468:
is the dimension of the positive-definite subspace of
2063:
1935:
1913:
1893:
1861:
1832:
1812:
1788:
1756:
1733:
1687:
1658:
1614:
1561:
1495:
1475:
1455:
1449:
is the intersection form on the second cohomology of
1435:
1290:
1246:
1226:
1206:
1164:
1132:
1112:
1092:
1063:
1043:
1011:
980:
951:
931:
861:
838:
807:
783:
759:
715:
680:
654:
622:
580:
538:
518:
474:
438:
418:
378:
332:
228:
201:
181:
148:
111:
1601:{\displaystyle \pm \alpha \in H^{2}(X;\mathbb {Z} )}
2349:
2069:
2057:from which one concludes the intersection form of
2046:
1919:
1899:
1879:
1847:
1818:
1794:
1774:
1739:
1719:
1673:
1644:
1600:
1547:
1481:
1461:
1441:
1418:
1274:
1232:
1212:
1192:
1150:
1118:
1098:
1078:
1049:
1029:
990:
963:
937:
913:
844:
824:
793:
769:
737:
690:
666:
640:
608:
566:
524:
504:
460:
424:
400:
360:
315:
207:
187:
166:
128:
1548:{\displaystyle c_{1}(L)\in H^{2}(X;\mathbb {Z} )}
2403:
219:, the dimension of the moduli space is given by
2114:2) Two smooth simply-connected 4-manifolds are
2367:
2380:
1802:itself at infinity. The resulting space is a
825:{\displaystyle {\mathcal {M}}_{\varepsilon }}
2350:Donaldson, S. K.; Kronheimer, P. B. (1990),
512:with respect to the intersection form. When
1489:are classified by their first Chern class
2325:
2309:
2228:
2210:
2187:
2169:
2016:
2013:
1887:(of unknown orientations). The signature
1867:
1864:
1762:
1759:
1720:{\displaystyle n(Q)\leq {\text{rank}}(Q)}
1591:
1538:
1066:
1017:
1014:
495:
700:
90:. This was a contribution cited for his
2389:
2404:
2262:
2260:
2258:
2256:
641:{\displaystyle \operatorname {SU} (2)}
167:{\displaystyle \operatorname {SU} (2)}
2292:
2290:
1645:{\displaystyle Q(\alpha ,\alpha )=-1}
505:{\displaystyle H_{2}(X,\mathbb {R} )}
66:(negative identity matrix) over the
2253:
13:
2354:, Oxford Mathematical Monographs,
2287:
2090:unimodular symmetric bilinear form
1126:by the standard representation of
983:
865:
811:
786:
762:
683:
237:
129:{\displaystyle {\mathcal {M}}_{P}}
115:
14:
2433:
2096:. Combining this result with the
1880:{\displaystyle \mathbb {CP} ^{2}}
1775:{\displaystyle \mathbb {CP} ^{2}}
1030:{\displaystyle \mathbb {CP} ^{2}}
2381:Freedman, M.; Quinn, F. (1990),
2313:Journal of Differential Geometry
2217:Journal of Differential Geometry
2176:Journal of Differential Geometry
1275:{\displaystyle E=L\oplus L^{-1}}
1193:{\displaystyle E=L\oplus L^{-1}}
1079:{\displaystyle \mathbb {C} ^{2}}
97:
2211:Donaldson, S. K. (1987-01-01).
2170:Donaldson, S. K. (1983-01-01).
1727:, with equality if and only if
886:
882:
102:Donaldson's proof utilizes the
2352:The Geometry of Four-Manifolds
2278:
2269:
2204:
2163:
2088:had previously shown that any
2041:
2035:
2026:
2008:
2002:
1993:
1984:
1978:
1969:
1963:
1947:
1941:
1842:
1836:
1782:. Secondly, glue in a copy of
1714:
1708:
1697:
1691:
1668:
1662:
1630:
1618:
1595:
1581:
1542:
1528:
1512:
1506:
1413:
1410:
1404:
1388:
1382:
1369:
1357:
1335:
1319:
1313:
1220:is a complex line bundle over
1145:
1139:
991:{\displaystyle {\mathcal {M}}}
908:
896:
794:{\displaystyle {\mathcal {M}}}
770:{\displaystyle {\mathcal {M}}}
732:
726:
691:{\displaystyle {\mathcal {M}}}
635:
629:
597:
591:
555:
549:
499:
485:
455:
449:
395:
389:
355:
349:
307:
304:
298:
282:
276:
257:
161:
155:
1:
2396:American Mathematical Society
2392:The Wild World of 4-Manifolds
2374:Instantons and Four-Manifolds
2303:
2080:
1652:. Let the number of pairs be
674:, one obtains a moduli space
2385:, Princeton University Press
2098:Serre classification theorem
852:, there is a diffeomorphism
845:{\displaystyle \varepsilon }
616:. Thus taking any principal
7:
2129:
217:Atiyah–Singer index theorem
138:anti-self-duality equations
10:
2438:
1469:. Since line bundles over
609:{\displaystyle b_{+}(X)=0}
567:{\displaystyle b_{1}(X)=0}
361:{\displaystyle k=c_{2}(P)}
86:The theorem was proved by
81:
2111:(so cannot be smoothed).
2156:
2109:differentiable structure
1826:and a disjoint union of
1004:complex projective plane
738:{\displaystyle b_{2}(X)}
461:{\displaystyle b_{+}(X)}
401:{\displaystyle b_{1}(X)}
2383:Topology of 4-Manifolds
1900:{\displaystyle \sigma }
195:over the four-manifold
2327:10.4310/jdg/1214437665
2230:10.4310/jdg/1214441485
2189:10.4310/jdg/1214437665
2071:
2048:
1921:
1901:
1881:
1849:
1820:
1796:
1776:
1741:
1721:
1675:
1646:
1602:
1549:
1483:
1463:
1443:
1420:
1276:
1234:
1214:
1194:
1152:
1120:
1100:
1080:
1051:
1031:
992:
965:
964:{\displaystyle x\in X}
939:
915:
846:
826:
795:
771:
739:
706:
692:
668:
642:
610:
568:
526:
506:
462:
426:
402:
362:
317:
209:
189:
168:
130:
2412:Differential topology
2072:
2049:
1922:
1902:
1882:
1850:
1821:
1797:
1777:
1742:
1722:
1676:
1647:
1603:
1550:
1484:
1464:
1444:
1421:
1277:
1235:
1215:
1195:
1153:
1151:{\displaystyle SU(2)}
1121:
1101:
1081:
1052:
1032:
993:
966:
940:
916:
847:
827:
796:
772:
740:
704:
693:
669:
643:
611:
569:
527:
507:
463:
427:
403:
363:
318:
210:
190:
169:
131:
25:differential topology
2417:Theorems in topology
2390:Scorpan, A. (2005),
2146:Yang–Mills equations
2105:topological manifold
2061:
1933:
1911:
1891:
1859:
1848:{\displaystyle n(Q)}
1830:
1810:
1786:
1754:
1731:
1685:
1674:{\displaystyle n(Q)}
1656:
1612:
1559:
1493:
1473:
1453:
1433:
1288:
1244:
1224:
1204:
1162:
1130:
1110:
1090:
1061:
1041:
1009:
978:
949:
929:
859:
836:
805:
781:
757:
713:
698:of dimension five.
678:
652:
620:
578:
536:
516:
472:
436:
416:
376:
330:
226:
199:
179:
146:
136:of solutions to the
109:
2077:is diagonalizable.
1747:is diagonalizable.
887:
667:{\displaystyle k=1}
33:Donaldson's theorem
2136:Unimodular lattice
2067:
2044:
1917:
1897:
1877:
1845:
1816:
1792:
1772:
1737:
1717:
1671:
1642:
1598:
1545:
1479:
1459:
1439:
1416:
1272:
1230:
1210:
1190:
1148:
1116:
1096:
1076:
1047:
1027:
988:
961:
935:
911:
842:
822:
791:
767:
735:
707:
688:
664:
638:
606:
564:
522:
502:
458:
422:
398:
358:
313:
205:
185:
164:
126:
2151:Rokhlin's theorem
2070:{\displaystyle X}
1939:
1920:{\displaystyle X}
1819:{\displaystyle X}
1795:{\displaystyle X}
1740:{\displaystyle Q}
1706:
1482:{\displaystyle X}
1462:{\displaystyle X}
1442:{\displaystyle Q}
1233:{\displaystyle X}
1213:{\displaystyle L}
1119:{\displaystyle P}
1099:{\displaystyle X}
1050:{\displaystyle E}
938:{\displaystyle X}
888:
753:show that whilst
745:many. Results of
525:{\displaystyle X}
425:{\displaystyle X}
208:{\displaystyle X}
188:{\displaystyle P}
76:fundamental group
40:intersection form
23:, and especially
2429:
2398:
2386:
2377:
2364:
2346:
2329:
2297:
2294:
2285:
2282:
2276:
2273:
2267:
2264:
2251:
2250:
2232:
2208:
2202:
2201:
2191:
2167:
2141:Donaldson theory
2086:Michael Freedman
2076:
2074:
2073:
2068:
2053:
2051:
2050:
2045:
2025:
2024:
2019:
1962:
1961:
1940:
1937:
1926:
1924:
1923:
1918:
1906:
1904:
1903:
1898:
1886:
1884:
1883:
1878:
1876:
1875:
1870:
1854:
1852:
1851:
1846:
1825:
1823:
1822:
1817:
1801:
1799:
1798:
1793:
1781:
1779:
1778:
1773:
1771:
1770:
1765:
1746:
1744:
1743:
1738:
1726:
1724:
1723:
1718:
1707:
1704:
1680:
1678:
1677:
1672:
1651:
1649:
1648:
1643:
1607:
1605:
1604:
1599:
1594:
1580:
1579:
1554:
1552:
1551:
1546:
1541:
1527:
1526:
1505:
1504:
1488:
1486:
1485:
1480:
1468:
1466:
1465:
1460:
1448:
1446:
1445:
1440:
1425:
1423:
1422:
1417:
1403:
1402:
1381:
1380:
1356:
1355:
1334:
1333:
1312:
1311:
1282:we may compute:
1281:
1279:
1278:
1273:
1271:
1270:
1239:
1237:
1236:
1231:
1219:
1217:
1216:
1211:
1199:
1197:
1196:
1191:
1189:
1188:
1157:
1155:
1154:
1149:
1125:
1123:
1122:
1117:
1105:
1103:
1102:
1097:
1085:
1083:
1082:
1077:
1075:
1074:
1069:
1056:
1054:
1053:
1048:
1036:
1034:
1033:
1028:
1026:
1025:
1020:
997:
995:
994:
989:
987:
986:
970:
968:
967:
962:
944:
942:
941:
936:
920:
918:
917:
912:
889:
877:
875:
874:
869:
868:
851:
849:
848:
843:
831:
829:
828:
823:
821:
820:
815:
814:
800:
798:
797:
792:
790:
789:
776:
774:
773:
768:
766:
765:
744:
742:
741:
736:
725:
724:
697:
695:
694:
689:
687:
686:
673:
671:
670:
665:
647:
645:
644:
639:
615:
613:
612:
607:
590:
589:
573:
571:
570:
565:
548:
547:
531:
529:
528:
523:
511:
509:
508:
503:
498:
484:
483:
467:
465:
464:
459:
448:
447:
431:
429:
428:
423:
407:
405:
404:
399:
388:
387:
367:
365:
364:
359:
348:
347:
322:
320:
319:
314:
297:
296:
275:
274:
241:
240:
214:
212:
211:
206:
194:
192:
191:
186:
173:
171:
170:
165:
135:
133:
132:
127:
125:
124:
119:
118:
72:simply connected
2437:
2436:
2432:
2431:
2430:
2428:
2427:
2426:
2422:Quadratic forms
2402:
2401:
2362:
2306:
2301:
2300:
2295:
2288:
2283:
2279:
2274:
2270:
2265:
2254:
2209:
2205:
2168:
2164:
2159:
2132:
2083:
2062:
2059:
2058:
2020:
2012:
2011:
1957:
1953:
1936:
1934:
1931:
1930:
1912:
1909:
1908:
1892:
1889:
1888:
1871:
1863:
1862:
1860:
1857:
1856:
1831:
1828:
1827:
1811:
1808:
1807:
1787:
1784:
1783:
1766:
1758:
1757:
1755:
1752:
1751:
1732:
1729:
1728:
1703:
1686:
1683:
1682:
1657:
1654:
1653:
1613:
1610:
1609:
1590:
1575:
1571:
1560:
1557:
1556:
1537:
1522:
1518:
1500:
1496:
1494:
1491:
1490:
1474:
1471:
1470:
1454:
1451:
1450:
1434:
1431:
1430:
1398:
1394:
1376:
1372:
1348:
1344:
1329:
1325:
1307:
1303:
1289:
1286:
1285:
1263:
1259:
1245:
1242:
1241:
1225:
1222:
1221:
1205:
1202:
1201:
1181:
1177:
1163:
1160:
1159:
1131:
1128:
1127:
1111:
1108:
1107:
1091:
1088:
1087:
1070:
1065:
1064:
1062:
1059:
1058:
1042:
1039:
1038:
1021:
1013:
1012:
1010:
1007:
1006:
982:
981:
979:
976:
975:
950:
947:
946:
930:
927:
926:
876:
870:
864:
863:
862:
860:
857:
856:
837:
834:
833:
816:
810:
809:
808:
806:
803:
802:
785:
784:
782:
779:
778:
761:
760:
758:
755:
754:
751:Karen Uhlenbeck
747:Clifford Taubes
720:
716:
714:
711:
710:
682:
681:
679:
676:
675:
653:
650:
649:
621:
618:
617:
585:
581:
579:
576:
575:
543:
539:
537:
534:
533:
517:
514:
513:
494:
479:
475:
473:
470:
469:
443:
439:
437:
434:
433:
417:
414:
413:
383:
379:
377:
374:
373:
343:
339:
331:
328:
327:
292:
288:
270:
266:
236:
235:
227:
224:
223:
200:
197:
196:
180:
177:
176:
147:
144:
143:
120:
114:
113:
112:
110:
107:
106:
100:
88:Simon Donaldson
84:
64:identity matrix
52:smooth manifold
17:
12:
11:
5:
2435:
2425:
2424:
2419:
2414:
2400:
2399:
2387:
2378:
2368:Freed, D. S.;
2365:
2360:
2347:
2320:(2): 279–315,
2305:
2302:
2299:
2298:
2286:
2277:
2268:
2252:
2203:
2161:
2160:
2158:
2155:
2154:
2153:
2148:
2143:
2138:
2131:
2128:
2126:, and parity.
2082:
2079:
2066:
2043:
2040:
2037:
2034:
2031:
2028:
2023:
2018:
2015:
2010:
2007:
2004:
2001:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1977:
1974:
1971:
1968:
1965:
1960:
1956:
1952:
1949:
1946:
1943:
1916:
1896:
1874:
1869:
1866:
1844:
1841:
1838:
1835:
1815:
1791:
1769:
1764:
1761:
1736:
1716:
1713:
1710:
1702:
1699:
1696:
1693:
1690:
1670:
1667:
1664:
1661:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1620:
1617:
1597:
1593:
1589:
1586:
1583:
1578:
1574:
1570:
1567:
1564:
1544:
1540:
1536:
1533:
1530:
1525:
1521:
1517:
1514:
1511:
1508:
1503:
1499:
1478:
1458:
1438:
1415:
1412:
1409:
1406:
1401:
1397:
1393:
1390:
1387:
1384:
1379:
1375:
1371:
1368:
1365:
1362:
1359:
1354:
1351:
1347:
1343:
1340:
1337:
1332:
1328:
1324:
1321:
1318:
1315:
1310:
1306:
1302:
1299:
1296:
1293:
1269:
1266:
1262:
1258:
1255:
1252:
1249:
1229:
1209:
1187:
1184:
1180:
1176:
1173:
1170:
1167:
1147:
1144:
1141:
1138:
1135:
1115:
1106:associated to
1095:
1073:
1068:
1046:
1024:
1019:
1016:
985:
960:
957:
954:
934:
923:
922:
910:
907:
904:
901:
898:
895:
892:
885:
880:
873:
867:
841:
819:
813:
788:
764:
734:
731:
728:
723:
719:
685:
663:
660:
657:
637:
634:
631:
628:
625:
605:
602:
599:
596:
593:
588:
584:
563:
560:
557:
554:
551:
546:
542:
521:
501:
497:
493:
490:
487:
482:
478:
457:
454:
451:
446:
442:
421:
397:
394:
391:
386:
382:
357:
354:
351:
346:
342:
338:
335:
324:
323:
312:
309:
306:
303:
300:
295:
291:
287:
284:
281:
278:
273:
269:
265:
262:
259:
256:
253:
250:
247:
244:
239:
234:
231:
204:
184:
163:
160:
157:
154:
151:
123:
117:
99:
96:
83:
80:
69:
60:diagonalizable
35:states that a
15:
9:
6:
4:
3:
2:
2434:
2423:
2420:
2418:
2415:
2413:
2410:
2409:
2407:
2397:
2393:
2388:
2384:
2379:
2375:
2371:
2370:Uhlenbeck, K.
2366:
2363:
2361:0-19-850269-9
2357:
2353:
2348:
2345:
2341:
2337:
2333:
2328:
2323:
2319:
2315:
2314:
2308:
2307:
2293:
2291:
2281:
2272:
2263:
2261:
2259:
2257:
2248:
2244:
2240:
2236:
2231:
2226:
2222:
2218:
2214:
2207:
2199:
2195:
2190:
2185:
2181:
2177:
2173:
2166:
2162:
2152:
2149:
2147:
2144:
2142:
2139:
2137:
2134:
2133:
2127:
2125:
2121:
2117:
2112:
2110:
2106:
2101:
2099:
2095:
2094:four-manifold
2091:
2087:
2078:
2064:
2055:
2038:
2032:
2029:
2021:
2005:
1999:
1996:
1990:
1987:
1981:
1975:
1972:
1966:
1958:
1954:
1950:
1944:
1928:
1927:is definite:
1914:
1894:
1872:
1839:
1833:
1813:
1805:
1789:
1767:
1748:
1734:
1711:
1700:
1694:
1688:
1665:
1659:
1639:
1636:
1633:
1627:
1624:
1621:
1615:
1587:
1584:
1576:
1572:
1568:
1565:
1562:
1534:
1531:
1523:
1519:
1515:
1509:
1501:
1497:
1476:
1456:
1436:
1427:
1407:
1399:
1395:
1391:
1385:
1377:
1373:
1366:
1363:
1360:
1352:
1349:
1345:
1341:
1338:
1330:
1326:
1322:
1316:
1308:
1304:
1300:
1297:
1294:
1291:
1283:
1267:
1264:
1260:
1256:
1253:
1250:
1247:
1227:
1207:
1185:
1182:
1178:
1174:
1171:
1168:
1165:
1142:
1136:
1133:
1113:
1093:
1086:-bundle over
1071:
1044:
1022:
1005:
1001:
972:
958:
955:
952:
932:
905:
902:
899:
893:
890:
883:
878:
871:
855:
854:
853:
839:
817:
752:
748:
729:
721:
717:
703:
699:
661:
658:
655:
648:-bundle with
632:
626:
623:
603:
600:
594:
586:
582:
561:
558:
552:
544:
540:
519:
491:
488:
480:
476:
452:
444:
440:
419:
411:
408:is the first
392:
384:
380:
371:
352:
344:
340:
336:
333:
310:
301:
293:
289:
285:
279:
271:
267:
263:
260:
254:
251:
248:
245:
242:
232:
229:
222:
221:
220:
218:
202:
182:
175:
158:
152:
149:
139:
121:
105:
98:Idea of proof
95:
93:
89:
79:
77:
73:
67:
65:
61:
57:
53:
49:
45:
41:
38:
34:
30:
26:
22:
2391:
2382:
2373:
2351:
2317:
2311:
2280:
2271:
2220:
2216:
2206:
2179:
2175:
2165:
2116:homeomorphic
2113:
2102:
2084:
2056:
1929:
1749:
1428:
1284:
973:
924:
708:
410:Betti number
325:
104:moduli space
101:
92:Fields medal
85:
32:
29:gauge theory
18:
1240:. Whenever
370:Chern class
21:mathematics
2406:Categories
2376:, Springer
2344:0507.57010
2304:References
2081:Extensions
1855:copies of
1608:such that
142:principal
2247:120208733
2239:0022-040X
2198:0022-040X
2124:signature
2030:≤
1997:⨆
1991:σ
1976:σ
1895:σ
1804:cobordism
1701:≤
1637:−
1628:α
1622:α
1569:∈
1566:α
1563:±
1516:∈
1364:−
1350:−
1342:⊕
1265:−
1257:⊕
1183:−
1175:⊕
1002:over the
956:∈
906:ε
894:×
884:≅
872:ε
840:ε
818:ε
627:
264:−
252:−
233:
215:. By the
153:
94:in 1986.
56:dimension
2372:(1984),
2130:See also
2107:with no
1806:between
879:→
68:integers
48:oriented
37:definite
2336:0710056
1057:be the
174:-bundle
82:History
44:compact
2358:
2342:
2334:
2245:
2237:
2196:
1429:where
1200:where
801:, say
432:, and
326:where
2243:S2CID
2223:(3).
2182:(2).
2157:Notes
1000:cones
368:is a
140:on a
58:4 is
42:of a
2356:ISBN
2235:ISSN
2194:ISSN
2120:rank
1938:rank
1705:rank
749:and
574:and
27:and
2340:Zbl
2322:doi
2225:doi
2184:doi
412:of
230:dim
54:of
19:In
2408::
2394:,
2338:,
2332:MR
2330:,
2318:18
2316:,
2289:^
2255:^
2241:.
2233:.
2221:26
2219:.
2215:.
2192:.
2180:18
2178:.
2174:.
2122:,
2054:,
1426:,
624:SU
372:,
150:SU
78:.
50:,
46:,
31:,
2324::
2249:.
2227::
2200:.
2186::
2065:X
2042:)
2039:Q
2036:(
2033:n
2027:)
2022:2
2017:P
2014:C
2009:)
2006:Q
2003:(
2000:n
1994:(
1988:=
1985:)
1982:X
1979:(
1973:=
1970:)
1967:X
1964:(
1959:2
1955:b
1951:=
1948:)
1945:Q
1942:(
1915:X
1873:2
1868:P
1865:C
1843:)
1840:Q
1837:(
1834:n
1814:X
1790:X
1768:2
1763:P
1760:C
1735:Q
1715:)
1712:Q
1709:(
1698:)
1695:Q
1692:(
1689:n
1669:)
1666:Q
1663:(
1660:n
1640:1
1634:=
1631:)
1625:,
1619:(
1616:Q
1596:)
1592:Z
1588:;
1585:X
1582:(
1577:2
1573:H
1543:)
1539:Z
1535:;
1532:X
1529:(
1524:2
1520:H
1513:)
1510:L
1507:(
1502:1
1498:c
1477:X
1457:X
1437:Q
1414:)
1411:)
1408:L
1405:(
1400:1
1396:c
1392:,
1389:)
1386:L
1383:(
1378:1
1374:c
1370:(
1367:Q
1361:=
1358:)
1353:1
1346:L
1339:L
1336:(
1331:2
1327:c
1323:=
1320:)
1317:E
1314:(
1309:2
1305:c
1301:=
1298:k
1295:=
1292:1
1268:1
1261:L
1254:L
1251:=
1248:E
1228:X
1208:L
1186:1
1179:L
1172:L
1169:=
1166:E
1146:)
1143:2
1140:(
1137:U
1134:S
1114:P
1094:X
1072:2
1067:C
1045:E
1023:2
1018:P
1015:C
984:M
959:X
953:x
933:X
921:.
909:)
903:,
900:0
897:(
891:X
866:M
812:M
787:M
763:M
733:)
730:X
727:(
722:2
718:b
684:M
662:1
659:=
656:k
636:)
633:2
630:(
604:0
601:=
598:)
595:X
592:(
587:+
583:b
562:0
559:=
556:)
553:X
550:(
545:1
541:b
520:X
500:)
496:R
492:,
489:X
486:(
481:2
477:H
456:)
453:X
450:(
445:+
441:b
420:X
396:)
393:X
390:(
385:1
381:b
356:)
353:P
350:(
345:2
341:c
337:=
334:k
311:,
308:)
305:)
302:X
299:(
294:+
290:b
286:+
283:)
280:X
277:(
272:1
268:b
261:1
258:(
255:3
249:k
246:8
243:=
238:M
203:X
183:P
162:)
159:2
156:(
122:P
116:M
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