1108:
representing the characteristic classes, take a wedge product so that one obtains a top dimensional form, then integrate over the manifold; this is analogous to taking the product in cohomology and pairing with the fundamental class.
947:
1721:
1244:
When the theory was put on an organised basis around 1950 (with the definitions reduced to homotopy theory) it became clear that the most fundamental characteristic classes known at that time (the
342:
849:
1396:
This is still the classic explanation, though in a given geometric theory it is profitable to take extra structure into account. When cohomology became 'extraordinary' with the arrival of
1180:
1145:
749:
798:
698:
1541:
1671:
267:
1835:
648:
1351:
1075:
1006:
1787:
1387:
487:
148:
1868:
1615:
1039:
970:
451:
1577:
1095:
414:
387:
207:
1966:
The appendix of this book: "Geometry of characteristic classes" is a very neat and profound introduction to the development of the ideas of characteristic classes.
1744:
180:
112:
1353:
was calculated, once and for all, the contravariance property of cohomology meant that characteristic classes for the bundle would be defined in
857:
1213:
a space, and to lead to a contradiction from the existence of a section we do need that variance. In fact cohomology theory grew up after
1202:
562:
573:
1676:
1269:
1189:: two manifolds are (respectively oriented or unoriented) cobordant if and only if their characteristic numbers are equal.
568:
Characteristic classes are elements of cohomology groups; one can obtain integers from characteristic classes, called
276:
2031:
1960:
1949:
803:
2050:
1419:; they have (in a modified sense, for foliations with some allowed singularities) a classifying space theory in
1150:
1115:
1997:
976:
of cohomology classes. These are notated variously as either the product of characteristic classes, such as
719:
1233:) was one major reason why a 'dual' theory to homology was sought. The characteristic class approach to
757:
657:
2019:
1496:
1272:. On the other hand there was now a framework which produced families of classes, whenever there was a
585:
1620:
1042:
581:
55:
product structure from a global product structure. They are one of the unifying geometric concepts in
2023:
1471:
1245:
1206:
43:. The cohomology class measures the extent to which the bundle is "twisted" and whether it possesses
1749:
This is not the case for the Euler class, as detailed there, not least because the Euler class of a
240:
1238:
1792:
605:
1723:
and similar). Equivalently, all finite characteristic classes pull back from a stable class in
1463:
1317:
1302:. For the homotopy theory the relevant information is carried by compact subgroups such as the
1214:
1048:
979:
359:
234:
218:
44:
1756:
1356:
456:
117:
1917:
1840:
1582:
1260:
structure. What is more, the Chern class itself was not so new, having been reflected in the
1011:
955:
210:
60:
52:
48:
430:
1884:
1550:
1080:
704:
with the fundamental class. The number of distinct characteristic numbers is the number of
392:
365:
185:
8:
222:
1976:
1726:
1493:
Concretely, a stable class is one that does not change when one adds a trivial bundle:
1230:
1186:
1101:
577:
165:
97:
64:
56:
2027:
2011:
1993:
1956:
1945:
1937:
1544:
1447:
1292:
1261:
1198:
1105:
600:
230:
151:
91:
1475:
1451:
1439:
1401:
1303:
1253:
155:
28:
1985:
1435:
1229:
a space; and characteristic class theory in its infancy in the 1930s (as part of
1218:
270:
75:
71:
2044:
1972:
1307:
1273:
1257:
1921:
1265:
2007:
1889:
1879:
1483:
1467:
1390:
1249:
973:
20:
1237:
invariants was a particular reason to make a theory, to prove a general
942:{\displaystyle c_{i_{1}}\smile c_{i_{2}}\smile \dots \smile c_{i_{l}}()}
1284:
1222:
36:
1443:
1412:
1404:
from 1955 onwards, it was really only necessary to change the letter
1234:
1182:-valued characteristic numbers, such as the Stiefel-Whitney numbers.
423:
In other words, a characteristic class associates to each principal
1420:
1416:
1397:
1393:
is really one class with graded components in each even dimension.
705:
700:, one can pair a product of characteristic classes of total degree
1431:
712:
in the characteristic classes, or equivalently the partitions of
70:
The notion of characteristic class arose in 1935 in the work of
1543:. More abstractly, it means that the cohomology class in the
1279:
The prime mechanism then appeared to be this: Given a space
1256:) were reflections of the classical linear groups and their
561:"Characteristic number" redirects here. For other uses, see
1112:
This also works for non-orientable manifolds, which have a
1185:
Characteristic numbers solve the oriented and unoriented
1907:
Informally, characteristic classes "live" in cohomology.
1408:
everywhere to say what the characteristic classes were.
572:. Some important examples of characteristic numbers are
16:
Association of cohomology classes to principal bundles
1716:{\displaystyle \mathbf {R} ^{n}\to \mathbf {R} ^{n+1}}
811:
724:
1843:
1795:
1759:
1729:
1679:
1623:
1585:
1553:
1499:
1359:
1320:
1153:
1118:
1083:
1051:
1014:
982:
958:
860:
806:
760:
722:
660:
608:
459:
433:
395:
368:
279:
243:
188:
168:
120:
100:
1992:(3rd Edition, Springer 1993 ed.). McGraw Hill.
1862:
1829:
1781:
1738:
1715:
1665:
1609:
1571:
1535:
1381:
1345:
1174:
1139:
1089:
1069:
1033:
1000:
964:
941:
843:
792:
743:
692:
642:
481:
445:
408:
381:
336:
261:
201:
174:
142:
106:
2042:
337:{\displaystyle f^{*}\colon b_{G}(Y)\to b_{G}(X)}
2018:. Annals of Mathematics Studies. Vol. 76.
541:). On the left is the class of the pullback of
2006:
1283:carrying a vector bundle, that implied in the
851:, the corresponding characteristic number is:
844:{\displaystyle \sum {\mbox{deg}}\,c_{i_{j}}=n}
1920:, these are polynomials in the curvature; by
1411:Characteristic classes were later found for
549:; on the right is the image of the class of
1434:, new characteristic classes were found by
1008:, or by some alternative notation, such as
1984:
1942:Complex manifolds without potential theory
1175:{\displaystyle \mathbf {Z} /2\mathbf {Z} }
1140:{\displaystyle \mathbf {Z} /2\mathbf {Z} }
817:
730:
556:
1837:, so it can't pull back from a class in
1579:pulls back from the cohomology class in
1389:in the same dimensions. For example the
1197:Characteristic classes are phenomena of
1147:-orientation, in which case one obtains
2043:
1446:theory. The work and point of view of
563:Characteristic number (disambiguation)
1936:
1201:in an essential way — they are
553:under the induced map in cohomology.
1673:(which corresponds to the inclusion
1270:Italian school of algebraic geometry
654:-bundle with characteristic classes
47:. Characteristic classes are global
1971:
744:{\displaystyle {\mbox{deg}}\,c_{i}}
13:
793:{\displaystyle i_{1},\dots ,i_{l}}
693:{\displaystyle c_{1},\dots ,c_{k}}
78:about vector fields on manifolds.
14:
2062:
1536:{\displaystyle c(V\oplus 1)=c(V)}
1205:constructions, in the way that a
1697:
1682:
1666:{\displaystyle BG(n)\to BG(n+1)}
1450:have also proved important: see
1298:, for the relevant linear group
1168:
1155:
1133:
1120:
416:, regarded also as a functor to
51:that measure the deviation of a
27:is a way of associating to each
1910:
1901:
1824:
1821:
1815:
1806:
1776:
1770:
1692:
1660:
1648:
1639:
1636:
1630:
1604:
1592:
1566:
1560:
1530:
1524:
1515:
1503:
1376:
1370:
1340:
1331:
1097:for the Euler characteristic.
936:
933:
927:
924:
637:
631:
615:
609:
476:
470:
437:
331:
325:
312:
309:
303:
262:{\displaystyle f\colon X\to Y}
253:
137:
131:
94:, and for a topological space
1:
1978:Vector bundles & K-theory
1930:
1924:, one can take harmonic form.
1753:-dimensional bundle lives in
1192:
81:
1870:, as the dimensions differ.
1830:{\displaystyle H^{k}(BO(k))}
1457:
643:{\displaystyle \in H_{n}(M)}
7:
1873:
591:Given an oriented manifold
10:
2067:
2020:Princeton University Press
1225:theories based on mapping
1100:From the point of view of
560:
517:is a continuous map, then
221:of topological spaces and
2024:University of Tokyo Press
1944:. Springer-Verlag Press.
1346:{\displaystyle H^{*}(BG)}
1070:{\displaystyle p_{1}^{2}}
1001:{\displaystyle c_{1}^{2}}
1895:
1782:{\displaystyle H^{k}(X)}
1426:In later work after the
1382:{\displaystyle H^{*}(X)}
482:{\displaystyle b_{G}(X)}
389:to a cohomology functor
143:{\displaystyle b_{G}(X)}
1863:{\displaystyle H^{k+1}}
1789:(hence pulls back from
1610:{\displaystyle BG(n+1)}
1034:{\displaystyle P_{1,1}}
965:{\displaystyle \smile }
574:Stiefel–Whitney numbers
2051:Characteristic classes
2016:Characteristic classes
1864:
1831:
1783:
1740:
1717:
1667:
1611:
1573:
1537:
1464:stable homotopy theory
1383:
1347:
1314:. Once the cohomology
1268:, and the work of the
1209:is a kind of function
1176:
1141:
1091:
1071:
1035:
1002:
966:
943:
845:
794:
745:
694:
644:
570:characteristic numbers
557:Characteristic numbers
483:
447:
446:{\displaystyle P\to X}
410:
383:
360:natural transformation
338:
263:
203:
176:
144:
108:
1865:
1832:
1784:
1741:
1718:
1668:
1612:
1574:
1572:{\displaystyle BG(n)}
1538:
1472:Stiefel–Whitney class
1384:
1348:
1246:Stiefel–Whitney class
1177:
1142:
1092:
1090:{\displaystyle \chi }
1072:
1036:
1003:
967:
944:
846:
795:
746:
695:
645:
484:
448:
411:
409:{\displaystyle H^{*}}
384:
382:{\displaystyle b_{G}}
339:
264:
211:contravariant functor
204:
202:{\displaystyle b_{G}}
177:
145:
109:
61:differential geometry
1885:Euler characteristic
1841:
1793:
1757:
1727:
1677:
1621:
1617:under the inclusion
1583:
1551:
1497:
1357:
1318:
1239:Gauss–Bonnet theorem
1151:
1116:
1081:
1049:
1012:
980:
956:
858:
804:
758:
720:
658:
606:
586:Euler characteristic
457:
431:
393:
366:
349:characteristic class
277:
241:
223:continuous functions
186:
166:
118:
98:
25:characteristic class
1462:In the language of
1452:Chern–Simons theory
1430:of mathematics and
1066:
997:
358:-bundles is then a
152:isomorphism classes
1938:Chern, Shiing-Shen
1860:
1827:
1779:
1739:{\displaystyle BG}
1736:
1713:
1663:
1607:
1569:
1533:
1379:
1343:
1254:Pontryagin classes
1231:obstruction theory
1172:
1137:
1106:differential forms
1102:de Rham cohomology
1087:
1067:
1052:
1031:
998:
983:
962:
939:
841:
815:
790:
741:
728:
690:
640:
582:Pontryagin numbers
479:
443:
406:
379:
334:
259:
199:
172:
140:
104:
65:algebraic geometry
57:algebraic topology
2022:, Princeton, NJ;
1918:Chern–Weil theory
1545:classifying space
1304:orthogonal groups
1293:classifying space
1285:homotopy category
1262:Schubert calculus
1221:, which are both
1199:cohomology theory
1187:bordism questions
1045:corresponding to
1043:Pontryagin number
814:
727:
601:fundamental class
237:), sending a map
229:(the category of
175:{\displaystyle X}
107:{\displaystyle X}
92:topological group
2058:
2037:
2003:
1986:Husemoller, Dale
1981:
1955:
1925:
1914:
1908:
1905:
1869:
1867:
1866:
1861:
1859:
1858:
1836:
1834:
1833:
1828:
1805:
1804:
1788:
1786:
1785:
1780:
1769:
1768:
1745:
1743:
1742:
1737:
1722:
1720:
1719:
1714:
1712:
1711:
1700:
1691:
1690:
1685:
1672:
1670:
1669:
1664:
1616:
1614:
1613:
1608:
1578:
1576:
1575:
1570:
1542:
1540:
1539:
1534:
1476:Pontryagin class
1440:Dieter Kotschick
1402:cobordism theory
1388:
1386:
1385:
1380:
1369:
1368:
1352:
1350:
1349:
1344:
1330:
1329:
1181:
1179:
1178:
1173:
1171:
1163:
1158:
1146:
1144:
1143:
1138:
1136:
1128:
1123:
1096:
1094:
1093:
1088:
1076:
1074:
1073:
1068:
1065:
1060:
1040:
1038:
1037:
1032:
1030:
1029:
1007:
1005:
1004:
999:
996:
991:
971:
969:
968:
963:
948:
946:
945:
940:
923:
922:
921:
920:
897:
896:
895:
894:
877:
876:
875:
874:
850:
848:
847:
842:
834:
833:
832:
831:
816:
812:
799:
797:
796:
791:
789:
788:
770:
769:
754:Formally, given
750:
748:
747:
742:
740:
739:
729:
725:
699:
697:
696:
691:
689:
688:
670:
669:
649:
647:
646:
641:
630:
629:
505:) such that, if
488:
486:
485:
480:
469:
468:
452:
450:
449:
444:
415:
413:
412:
407:
405:
404:
388:
386:
385:
380:
378:
377:
343:
341:
340:
335:
324:
323:
302:
301:
289:
288:
268:
266:
265:
260:
208:
206:
205:
200:
198:
197:
181:
179:
178:
173:
149:
147:
146:
141:
130:
129:
113:
111:
110:
105:
29:principal bundle
2066:
2065:
2061:
2060:
2059:
2057:
2056:
2055:
2041:
2040:
2034:
2008:Milnor, John W.
2000:
1952:
1933:
1928:
1915:
1911:
1906:
1902:
1898:
1876:
1848:
1844:
1842:
1839:
1838:
1800:
1796:
1794:
1791:
1790:
1764:
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1758:
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1754:
1728:
1725:
1724:
1701:
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1686:
1681:
1680:
1678:
1675:
1674:
1622:
1619:
1618:
1584:
1581:
1580:
1552:
1549:
1548:
1498:
1495:
1494:
1460:
1436:Simon Donaldson
1364:
1360:
1358:
1355:
1354:
1325:
1321:
1319:
1316:
1315:
1287:a mapping from
1219:homotopy theory
1195:
1167:
1159:
1154:
1152:
1149:
1148:
1132:
1124:
1119:
1117:
1114:
1113:
1104:, one can take
1082:
1079:
1078:
1061:
1056:
1050:
1047:
1046:
1019:
1015:
1013:
1010:
1009:
992:
987:
981:
978:
977:
957:
954:
953:
916:
912:
911:
907:
890:
886:
885:
881:
870:
866:
865:
861:
859:
856:
855:
827:
823:
822:
818:
810:
805:
802:
801:
784:
780:
765:
761:
759:
756:
755:
735:
731:
723:
721:
718:
717:
684:
680:
665:
661:
659:
656:
655:
625:
621:
607:
604:
603:
566:
559:
464:
460:
458:
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454:
432:
429:
428:
400:
396:
394:
391:
390:
373:
369:
367:
364:
363:
319:
315:
297:
293:
284:
280:
278:
275:
274:
242:
239:
238:
193:
189:
187:
184:
183:
167:
164:
163:
150:for the set of
125:
121:
119:
116:
115:
99:
96:
95:
84:
76:Hassler Whitney
17:
12:
11:
5:
2064:
2054:
2053:
2039:
2038:
2032:
2004:
1998:
1982:
1973:Hatcher, Allen
1969:
1968:
1967:
1950:
1932:
1929:
1927:
1926:
1909:
1899:
1897:
1894:
1893:
1892:
1887:
1882:
1875:
1872:
1857:
1854:
1851:
1847:
1826:
1823:
1820:
1817:
1814:
1811:
1808:
1803:
1799:
1778:
1775:
1772:
1767:
1763:
1735:
1732:
1710:
1707:
1704:
1699:
1694:
1689:
1684:
1662:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1632:
1629:
1626:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1568:
1565:
1562:
1559:
1556:
1532:
1529:
1526:
1523:
1520:
1517:
1514:
1511:
1508:
1505:
1502:
1459:
1456:
1378:
1375:
1372:
1367:
1363:
1342:
1339:
1336:
1333:
1328:
1324:
1308:unitary groups
1194:
1191:
1170:
1166:
1162:
1157:
1135:
1131:
1127:
1122:
1086:
1064:
1059:
1055:
1028:
1025:
1022:
1018:
995:
990:
986:
961:
950:
949:
938:
935:
932:
929:
926:
919:
915:
910:
906:
903:
900:
893:
889:
884:
880:
873:
869:
864:
840:
837:
830:
826:
821:
809:
787:
783:
779:
776:
773:
768:
764:
738:
734:
687:
683:
679:
676:
673:
668:
664:
639:
636:
633:
628:
624:
620:
617:
614:
611:
558:
555:
478:
475:
472:
467:
463:
442:
439:
436:
403:
399:
376:
372:
333:
330:
327:
322:
318:
314:
311:
308:
305:
300:
296:
292:
287:
283:
258:
255:
252:
249:
246:
196:
192:
171:
139:
136:
133:
128:
124:
103:
83:
80:
72:Eduard Stiefel
15:
9:
6:
4:
3:
2:
2063:
2052:
2049:
2048:
2046:
2035:
2033:0-691-08122-0
2029:
2025:
2021:
2017:
2013:
2012:Stasheff, Jim
2009:
2005:
2001:
1995:
1991:
1990:Fibre bundles
1987:
1983:
1980:
1979:
1974:
1970:
1965:
1964:
1962:
1961:3-540-90422-0
1958:
1953:
1951:0-387-90422-0
1947:
1943:
1939:
1935:
1934:
1923:
1919:
1913:
1904:
1900:
1891:
1888:
1886:
1883:
1881:
1878:
1877:
1871:
1855:
1852:
1849:
1845:
1818:
1812:
1809:
1801:
1797:
1773:
1765:
1761:
1752:
1747:
1733:
1730:
1708:
1705:
1702:
1687:
1657:
1654:
1651:
1645:
1642:
1633:
1627:
1624:
1601:
1598:
1595:
1589:
1586:
1563:
1557:
1554:
1546:
1527:
1521:
1518:
1512:
1509:
1506:
1500:
1491:
1489:
1485:
1481:
1477:
1473:
1469:
1465:
1455:
1453:
1449:
1445:
1441:
1437:
1433:
1429:
1428:rapprochement
1424:
1422:
1418:
1414:
1409:
1407:
1403:
1399:
1394:
1392:
1373:
1365:
1361:
1337:
1334:
1326:
1322:
1313:
1309:
1305:
1301:
1297:
1294:
1290:
1286:
1282:
1277:
1275:
1274:vector bundle
1271:
1267:
1266:Grassmannians
1263:
1259:
1258:maximal torus
1255:
1251:
1247:
1242:
1240:
1236:
1232:
1228:
1224:
1220:
1216:
1212:
1208:
1204:
1203:contravariant
1200:
1190:
1188:
1183:
1164:
1160:
1129:
1125:
1110:
1107:
1103:
1098:
1084:
1062:
1057:
1053:
1044:
1026:
1023:
1020:
1016:
993:
988:
984:
975:
959:
930:
917:
913:
908:
904:
901:
898:
891:
887:
882:
878:
871:
867:
862:
854:
853:
852:
838:
835:
828:
824:
819:
807:
785:
781:
777:
774:
771:
766:
762:
752:
736:
732:
715:
711:
707:
703:
685:
681:
677:
674:
671:
666:
662:
653:
634:
626:
622:
618:
612:
602:
598:
595:of dimension
594:
589:
587:
583:
579:
578:Chern numbers
575:
571:
564:
554:
552:
548:
544:
540:
536:
532:
528:
524:
520:
516:
512:
508:
504:
500:
496:
492:
473:
465:
461:
440:
434:
426:
421:
419:
401:
397:
374:
370:
361:
357:
354:of principal
353:
350:
345:
328:
320:
316:
306:
298:
294:
290:
285:
281:
272:
256:
250:
247:
244:
236:
232:
228:
224:
220:
216:
212:
194:
190:
169:
161:
159:
153:
134:
126:
122:
101:
93:
89:
79:
77:
73:
68:
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
2015:
1989:
1977:
1941:
1922:Hodge theory
1912:
1903:
1750:
1748:
1492:
1487:
1482:, while the
1479:
1461:
1427:
1425:
1410:
1405:
1395:
1311:
1299:
1295:
1288:
1280:
1278:
1243:
1226:
1210:
1196:
1184:
1111:
1099:
972:denotes the
951:
753:
713:
709:
701:
651:
596:
592:
590:
569:
567:
550:
546:
542:
538:
534:
530:
526:
522:
518:
514:
510:
506:
502:
498:
494:
490:
424:
422:
417:
355:
351:
348:
346:
226:
214:
157:
87:
85:
69:
40:
32:
24:
18:
1890:Chern class
1880:Segre class
1484:Euler class
1468:Chern class
1391:Chern class
1250:Chern class
974:cup product
489:an element
21:mathematics
1999:0387940871
1931:References
1413:foliations
1276:involved.
1252:, and the
1193:Motivation
800:such that
708:of degree
584:, and the
273:operation
156:principal
82:Definition
49:invariants
37:cohomology
2026:, Tokyo.
1693:→
1640:→
1510:⊕
1458:Stability
1444:instanton
1417:manifolds
1366:∗
1327:∗
1235:curvature
1223:covariant
1085:χ
960:⌣
905:⌣
902:⋯
899:⌣
879:⌣
808:∑
775:…
706:monomials
675:…
619:∈
438:→
402:∗
313:→
291::
286:∗
254:→
248::
235:functions
39:class of
2045:Category
2014:(1974).
1988:(1966).
1940:(1995).
1874:See also
1488:unstable
1423:theory.
1421:homotopy
1398:K-theory
1215:homology
1041:for the
650:, and a
509: :
427:-bundle
271:pullback
219:category
160:-bundles
114:, write
45:sections
1442:in the
1432:physics
1207:section
269:to the
182:. This
2030:
1996:
1959:
1948:
1480:stable
1474:, and
1466:, the
1248:, the
952:where
63:, and
1896:Notes
1448:Chern
1291:to a
1077:, or
716:into
599:with
497:) in
362:from
225:) to
217:(the
213:from
209:is a
162:over
90:be a
53:local
2028:ISBN
1994:ISBN
1957:ISBN
1946:ISBN
1547:for
1478:are
1438:and
1400:and
1306:and
1227:into
1217:and
529:) =
233:and
231:sets
86:Let
74:and
23:, a
1916:By
1486:is
1415:of
1310:of
1264:on
813:deg
726:deg
545:to
453:in
418:Set
227:Set
215:Top
154:of
31:of
19:In
2047::
2010:;
1975:,
1963:.
1746:.
1490:.
1470:,
1454:.
1296:BG
1241:.
1211:on
751:.
588:.
580:,
576:,
513:→
501:*(
420:.
347:A
344:.
67:.
59:,
35:a
2036:.
2002:.
1954:.
1856:1
1853:+
1850:k
1846:H
1825:)
1822:)
1819:k
1816:(
1813:O
1810:B
1807:(
1802:k
1798:H
1777:)
1774:X
1771:(
1766:k
1762:H
1751:k
1734:G
1731:B
1709:1
1706:+
1703:n
1698:R
1688:n
1683:R
1661:)
1658:1
1655:+
1652:n
1649:(
1646:G
1643:B
1637:)
1634:n
1631:(
1628:G
1625:B
1605:)
1602:1
1599:+
1596:n
1593:(
1590:G
1587:B
1567:)
1564:n
1561:(
1558:G
1555:B
1531:)
1528:V
1525:(
1522:c
1519:=
1516:)
1513:1
1507:V
1504:(
1501:c
1406:H
1377:)
1374:X
1371:(
1362:H
1341:)
1338:G
1335:B
1332:(
1323:H
1312:G
1300:G
1289:X
1281:X
1169:Z
1165:2
1161:/
1156:Z
1134:Z
1130:2
1126:/
1121:Z
1063:2
1058:1
1054:p
1027:1
1024:,
1021:1
1017:P
994:2
989:1
985:c
937:)
934:]
931:M
928:[
925:(
918:l
914:i
909:c
892:2
888:i
883:c
872:1
868:i
863:c
839:n
836:=
829:j
825:i
820:c
786:l
782:i
778:,
772:,
767:1
763:i
737:i
733:c
714:n
710:n
702:n
686:k
682:c
678:,
672:,
667:1
663:c
652:G
638:)
635:M
632:(
627:n
623:H
616:]
613:M
610:[
597:n
593:M
565:.
551:P
547:Y
543:P
539:P
537:(
535:c
533:*
531:f
527:P
525:*
523:f
521:(
519:c
515:X
511:Y
507:f
503:X
499:H
495:P
493:(
491:c
477:)
474:X
471:(
466:G
462:b
441:X
435:P
425:G
398:H
375:G
371:b
356:G
352:c
332:)
329:X
326:(
321:G
317:b
310:)
307:Y
304:(
299:G
295:b
282:f
257:Y
251:X
245:f
195:G
191:b
170:X
158:G
138:)
135:X
132:(
127:G
123:b
102:X
88:G
41:X
33:X
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