680:
493:
966:
675:{\displaystyle \cdots \longrightarrow H^{n}(E){\stackrel {\pi _{*}}{\longrightarrow }}H^{n-k}(M){\stackrel {e_{\wedge }}{\longrightarrow }}H^{n+1}(M){\stackrel {\pi ^{*}}{\longrightarrow }}H^{n+1}(E)\longrightarrow \cdots }
167:
358:
471:
848:
1059:
1093:
1170:
392:
281:
710:
251:
1343:
1240:
224:
112:
117:
1365:
1329:
961:{\displaystyle i^{!}:A_{k}(Y'){\overset {\sigma }{\longrightarrow }}A_{k}(N){\overset {\text{Gysin}}{\longrightarrow }}A_{k-d}(X')}
288:
399:
1432:
976:
1013:
477:
68:
1067:
1203:
1124:
1106:
1110:
32:
736:
with integral coefficients. In the integral case one needs to replace the wedge product with the
370:
259:
729:
688:
364:
188:
36:
24:
229:
1403:
1361:
1325:
1247:
765:
192:
1383:
1395:
1351:
1263:
1415:
1375:
1209:
1411:
1371:
1347:
209:
97:
52:
1064:
The second map is the (usual) Gysin homomorphism induced by the zero-section embedding
92:
1426:
1268:
713:
60:
48:
484:: it is a covariant map between objects associated with a contravariant functor.
988:
741:
737:
56:
44:
20:
1356:
1317:
733:
481:
40:
1407:
1384:"Zur Homologietheorie der Abbildungen und Faserungen von Mannigfaltigkeiten"
162:{\displaystyle S^{k}\hookrightarrow E{\stackrel {\pi }{\longrightarrow }}M.}
254:
1399:
744:, and the pushforward map no longer corresponds to integration.
728:
The Gysin sequence is a long exact sequence not only for the
353:{\displaystyle \pi ^{*}:H^{*}(M)\longrightarrow H^{*}(E).\,}
1346:. 3. Folge., vol. 2 (2nd ed.), Berlin, New York:
466:{\displaystyle \pi _{*}:H^{*}(E)\longrightarrow H^{*-k}(M)}
59:
of the sphere bundle and vice versa. It was introduced by
51:. The Gysin sequence is a useful tool for calculating the
487:
Gysin proved that the following is a long exact sequence
79:
Consider a fiber-oriented sphere bundle with total space
1212:
1127:
1113:
in that one either shows the intersection product of
1070:
1016:
851:
691:
496:
402:
373:
363:
In the case of a fiber bundle, one can also define a
291:
262:
232:
212:
120:
100:
1324:, Graduate Texts in Mathematics, Springer-Verlag,
1234:
1164:
1087:
1053:
960:
704:
674:
465:
386:
352:
275:
245:
218:
161:
106:
716:of a differential form with the Euler class
1344:Ergebnisse der Mathematik und ihrer Grenzgebiete
191:. There cohomology classes are represented by
8:
187:Discussion of the sequence is clearest with
478:fiberwise integration of differential forms
1054:{\displaystyle C_{X'/Y'}\hookrightarrow N}
1355:
1217:
1211:
1144:
1126:
1069:
1030:
1021:
1015:
932:
918:
903:
889:
869:
856:
850:
696:
690:
645:
631:
626:
621:
619:
618:
597:
583:
578:
573:
571:
570:
549:
535:
530:
525:
523:
522:
507:
495:
442:
420:
407:
401:
378:
372:
349:
331:
309:
296:
290:
267:
261:
237:
231:
211:
145:
140:
138:
137:
125:
119:
99:
1322:Differential Forms in Algebraic Topology
824:be the pullback of the normal bundle of
748:Gysin homomorphism in algebraic geometry
1280:
1172:or takes this formula as a definition.
1299:
1287:
179:called the Euler class of the bundle.
64:
7:
732:of differential forms, but also for
1088:{\displaystyle X'\hookrightarrow N}
480:on the oriented sphere – note that
1242:is the class of the zero-locus of
14:
1388:Commentarii Mathematici Helvetici
171:Any such bundle defines a degree
175: + 1 cohomology class
1165:{\displaystyle X\cdot V=i^{!},}
1229:
1223:
1156:
1150:
1079:
1045:
955:
944:
920:
915:
909:
891:
886:
875:
666:
663:
657:
622:
615:
609:
574:
567:
561:
526:
519:
513:
500:
460:
454:
435:
432:
426:
343:
337:
324:
321:
315:
141:
131:
1:
482:this map goes "the wrong way"
67:), and is generalized by the
387:{\displaystyle \pi _{\ast }}
276:{\displaystyle \pi ^{\ast }}
226:induces a map in cohomology
1121:to be given by the formula
977:specialization homomorphism
705:{\displaystyle e_{\wedge }}
1449:
842:refers to the composition
837:refined Gysin homomorphism
1357:10.1007/978-1-4612-1700-8
246:{\displaystyle H^{\ast }}
199:can be represented by a (
1338:Fulton, William (1998),
1302:, Proposition 14.1. (c).
1178:: Given a vector bundle
983:-dimensional subvariety
991:to the intersection of
69:Serre spectral sequence
1382:Gysin, Werner (1942),
1236:
1166:
1089:
1055:
962:
706:
676:
467:
388:
354:
277:
247:
220:
203: + 1)-form.
163:
108:
1320:; Tu, Loring (1982),
1237:
1235:{\displaystyle s^{!}}
1167:
1090:
1056:
1006:. The result lies in
963:
820:the induced map. Let
707:
677:
468:
389:
355:
278:
248:
221:
164:
109:
1210:
1125:
1107:intersection product
1068:
1014:
849:
689:
494:
400:
371:
289:
260:
230:
219:{\displaystyle \pi }
210:
118:
107:{\displaystyle \pi }
98:
43:, the fiber and the
1340:Intersection theory
1111:intersection theory
724:Integral cohomology
206:The projection map
33:long exact sequence
16:Long exact sequence
1433:Algebraic topology
1400:10.1007/bf02565612
1232:
1162:
1085:
1051:
958:
730:de Rham cohomology
702:
672:
463:
384:
350:
273:
243:
216:
193:differential forms
189:de Rham cohomology
183:De Rham cohomology
159:
104:
37:cohomology classes
35:which relates the
25:algebraic topology
1367:978-3-540-62046-4
1290:, Example 6.2.1..
1248:fundamental class
1099:The homomorphism
926:
925:
897:
766:regular embedding
638:
590:
542:
150:
1440:
1418:
1378:
1359:
1334:
1303:
1297:
1291:
1285:
1264:Logarithmic form
1246:, where is the
1241:
1239:
1238:
1233:
1222:
1221:
1194:be a section of
1171:
1169:
1168:
1163:
1149:
1148:
1094:
1092:
1091:
1086:
1078:
1060:
1058:
1057:
1052:
1044:
1043:
1042:
1034:
1029:
1000:
979:; which sends a
967:
965:
964:
959:
954:
943:
942:
927:
923:
919:
908:
907:
898:
890:
885:
874:
873:
861:
860:
833:
819:
812:
796:
788:
777:
711:
709:
708:
703:
701:
700:
681:
679:
678:
673:
656:
655:
640:
639:
637:
636:
635:
625:
620:
608:
607:
592:
591:
589:
588:
587:
577:
572:
560:
559:
544:
543:
541:
540:
539:
529:
524:
512:
511:
472:
470:
469:
464:
453:
452:
425:
424:
412:
411:
393:
391:
390:
385:
383:
382:
359:
357:
356:
351:
336:
335:
314:
313:
301:
300:
282:
280:
279:
274:
272:
271:
252:
250:
249:
244:
242:
241:
225:
223:
222:
217:
168:
166:
165:
160:
152:
151:
149:
144:
139:
130:
129:
113:
111:
110:
105:
53:cohomology rings
19:In the field of
1448:
1447:
1443:
1442:
1441:
1439:
1438:
1437:
1423:
1422:
1421:
1381:
1368:
1348:Springer-Verlag
1337:
1332:
1331:978-038790613-3
1316:
1312:
1307:
1306:
1298:
1294:
1286:
1282:
1277:
1260:
1213:
1208:
1207:
1204:regular section
1140:
1123:
1122:
1071:
1066:
1065:
1035:
1022:
1017:
1012:
1011:
998:
947:
928:
899:
878:
865:
852:
847:
846:
831:
817:
810:
806:
794:
786:
783:a morphism and
775:
768:of codimension
750:
726:
692:
687:
686:
641:
627:
593:
579:
545:
531:
503:
492:
491:
438:
416:
403:
398:
397:
374:
369:
368:
327:
305:
292:
287:
286:
263:
258:
257:
233:
228:
227:
208:
207:
185:
121:
116:
115:
96:
95:
77:
17:
12:
11:
5:
1446:
1444:
1436:
1435:
1425:
1424:
1420:
1419:
1379:
1366:
1335:
1330:
1313:
1311:
1308:
1305:
1304:
1292:
1279:
1278:
1276:
1273:
1272:
1271:
1266:
1259:
1256:
1231:
1228:
1225:
1220:
1216:
1161:
1158:
1155:
1152:
1147:
1143:
1139:
1136:
1133:
1130:
1097:
1096:
1084:
1081:
1077:
1074:
1062:
1050:
1047:
1041:
1038:
1033:
1028:
1025:
1020:
969:
968:
957:
953:
950:
946:
941:
938:
935:
931:
922:
917:
914:
911:
906:
902:
896:
893:
888:
884:
881:
877:
872:
868:
864:
859:
855:
802:
764:be a (closed)
749:
746:
725:
722:
699:
695:
683:
682:
671:
668:
665:
662:
659:
654:
651:
648:
644:
634:
630:
624:
617:
614:
611:
606:
603:
600:
596:
586:
582:
576:
569:
566:
563:
558:
555:
552:
548:
538:
534:
528:
521:
518:
515:
510:
506:
502:
499:
476:which acts by
474:
473:
462:
459:
456:
451:
448:
445:
441:
437:
434:
431:
428:
423:
419:
415:
410:
406:
381:
377:
361:
360:
348:
345:
342:
339:
334:
330:
326:
323:
320:
317:
312:
308:
304:
299:
295:
270:
266:
240:
236:
215:
184:
181:
158:
155:
148:
143:
136:
133:
128:
124:
103:
93:projection map
76:
73:
29:Gysin sequence
15:
13:
10:
9:
6:
4:
3:
2:
1445:
1434:
1431:
1430:
1428:
1417:
1413:
1409:
1405:
1401:
1397:
1393:
1389:
1385:
1380:
1377:
1373:
1369:
1363:
1358:
1353:
1349:
1345:
1341:
1336:
1333:
1327:
1323:
1319:
1315:
1314:
1309:
1301:
1296:
1293:
1289:
1284:
1281:
1274:
1270:
1269:Wang sequence
1267:
1265:
1262:
1261:
1257:
1255:
1253:
1249:
1245:
1226:
1218:
1214:
1205:
1201:
1198:. Then, when
1197:
1193:
1189:
1185:
1181:
1177:
1173:
1159:
1153:
1145:
1141:
1137:
1134:
1131:
1128:
1120:
1116:
1112:
1108:
1105:
1102:
1082:
1075:
1072:
1063:
1048:
1039:
1036:
1031:
1026:
1023:
1018:
1009:
1005:
1001:
994:
990:
986:
982:
978:
974:
973:
972:
951:
948:
939:
936:
933:
929:
912:
904:
900:
894:
882:
879:
870:
866:
862:
857:
853:
845:
844:
843:
841:
838:
834:
827:
823:
816:
809:
805:
800:
793:
789:
782:
778:
771:
767:
763:
759:
755:
747:
745:
743:
739:
735:
731:
723:
721:
719:
715:
714:wedge product
697:
693:
669:
660:
652:
649:
646:
642:
632:
628:
612:
604:
601:
598:
594:
584:
580:
564:
556:
553:
550:
546:
536:
532:
516:
508:
504:
497:
490:
489:
488:
485:
483:
479:
457:
449:
446:
443:
439:
429:
421:
417:
413:
408:
404:
396:
395:
394:
379:
375:
366:
346:
340:
332:
328:
318:
310:
306:
302:
297:
293:
285:
284:
283:
268:
264:
256:
238:
234:
213:
204:
202:
198:
194:
190:
182:
180:
178:
174:
169:
156:
153:
146:
134:
126:
122:
101:
94:
90:
86:
83:, base space
82:
74:
72:
70:
66:
62:
58:
54:
50:
49:sphere bundle
46:
42:
38:
34:
30:
26:
22:
1391:
1387:
1339:
1321:
1295:
1283:
1251:
1243:
1199:
1195:
1191:
1187:
1183:
1179:
1175:
1174:
1118:
1114:
1103:
1100:
1098:
1007:
1003:
996:
992:
984:
980:
970:
839:
836:
829:
825:
821:
814:
807:
803:
798:
791:
784:
780:
773:
769:
761:
757:
753:
751:
727:
717:
684:
486:
475:
362:
205:
200:
196:
186:
176:
172:
170:
88:
84:
80:
78:
28:
18:
1318:Bott, Raoul
1300:Fulton 1998
1288:Fulton 1998
989:normal cone
835:. Then the
742:cup product
738:Euler class
365:pushforward
253:called its
57:Euler class
45:total space
21:mathematics
1394:: 61–122,
734:cohomology
195:, so that
75:Definition
55:given the
41:base space
1408:0010-2571
1132:⋅
1080:↪
1046:↪
975:σ is the
937:−
921:⟶
895:σ
892:⟶
740:with the
698:∧
670:⋯
667:⟶
633:∗
629:π
623:⟶
585:∧
575:⟶
554:−
537:∗
533:π
527:⟶
501:⟶
498:⋯
447:−
444:∗
436:⟶
422:∗
409:∗
405:π
380:∗
376:π
333:∗
325:⟶
311:∗
298:∗
294:π
269:∗
265:π
239:∗
214:π
147:π
142:⟶
132:↪
102:π
23:known as
1427:Category
1258:See also
1076:′
1040:′
1027:′
1010:through
952:′
883:′
255:pullback
87:, fiber
1416:0006511
1376:1644323
1310:Sources
1176:Example
1104:encodes
987:to the
712:is the
63: (
39:of the
1414:
1406:
1374:
1364:
1328:
1182:, let
971:where
685:where
27:, the
1275:Notes
1202:is a
999:'
924:Gysin
832:'
818:'
811:'
795:'
787:'
776:'
61:Gysin
47:of a
31:is a
1404:ISSN
1362:ISBN
1326:ISBN
1117:and
995:and
752:Let
367:map
91:and
65:1942
1396:doi
1352:doi
1250:of
1109:in
1002:in
828:to
1429::
1412:MR
1410:,
1402:,
1392:14
1390:,
1386:,
1372:MR
1370:,
1360:,
1350:,
1342:,
1254:.
1206:,
1190:→
1186::
813:→
797:=
790::
779:→
772:,
760:→
756::
720:.
114::
71:.
1398::
1354::
1252:X
1244:s
1230:]
1227:X
1224:[
1219:!
1215:s
1200:s
1196:E
1192:E
1188:X
1184:s
1180:E
1160:,
1157:]
1154:V
1151:[
1146:!
1142:i
1138:=
1135:V
1129:X
1119:V
1115:X
1101:i
1095:.
1083:N
1073:X
1061:.
1049:N
1037:Y
1032:/
1024:X
1019:C
1008:N
1004:V
997:X
993:V
985:V
981:k
956:)
949:X
945:(
940:d
934:k
930:A
916:)
913:N
910:(
905:k
901:A
887:)
880:Y
876:(
871:k
867:A
863::
858:!
854:i
840:i
830:X
826:i
822:N
815:Y
808:Y
804:Y
801:×
799:X
792:X
785:i
781:Y
774:Y
770:d
762:Y
758:X
754:i
718:e
694:e
664:)
661:E
658:(
653:1
650:+
647:n
643:H
616:)
613:M
610:(
605:1
602:+
599:n
595:H
581:e
568:)
565:M
562:(
557:k
551:n
547:H
520:)
517:E
514:(
509:n
505:H
461:)
458:M
455:(
450:k
440:H
433:)
430:E
427:(
418:H
414::
347:.
344:)
341:E
338:(
329:H
322:)
319:M
316:(
307:H
303::
235:H
201:k
197:e
177:e
173:k
157:.
154:M
135:E
127:k
123:S
89:S
85:M
81:E
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