623:
1147:
892:
483:
751:
1261:
1006:
339:
238:
1340:
511:
1373:
1297:
1183:
1042:
928:
787:
659:
1049:
794:
385:
666:
1190:
935:
253:
136:
492:, which means the maximum number of graded elements of degree ≥ 1 that when multiplied give a non-zero result. For example a
375:
in the sense that the cup product commutes up to a sign determined by the grading. Specifically, for pure elements of degree
1438:
1417:
1313:
618:{\displaystyle \operatorname {H} ^{*}(\mathbb {R} P^{n};\mathbb {F} _{2})=\mathbb {F} _{2}/(\alpha ^{n+1})}
1349:
1457:
1142:{\displaystyle \operatorname {H} ^{*}(\mathbb {H} P^{n};\mathbb {Z} )=\mathbb {Z} /(\alpha ^{n+1})}
887:{\displaystyle \operatorname {H} ^{*}(\mathbb {C} P^{n};\mathbb {Z} )=\mathbb {Z} /(\alpha ^{n+1})}
493:
478:{\displaystyle (\alpha ^{k}\smile \beta ^{\ell })=(-1)^{k\ell }(\beta ^{\ell }\smile \alpha ^{k}).}
1378:
The reduced cohomology ring of wedge sums is the direct product of their reduced cohomology rings.
1266:
1152:
1011:
897:
756:
746:{\displaystyle \operatorname {H} ^{*}(\mathbb {R} P^{\infty };\mathbb {F} _{2})=\mathbb {F} _{2}}
628:
1382:
244:
8:
1256:{\displaystyle \operatorname {H} ^{*}(\mathbb {H} P^{\infty };\mathbb {Z} )=\mathbb {Z} }
1001:{\displaystyle \operatorname {H} ^{*}(\mathbb {C} P^{\infty };\mathbb {Z} )=\mathbb {Z} }
52:
1395:
372:
64:
56:
36:
21:
1428:
1303:
1434:
1413:
497:
68:
29:
91:
51:
serving as the ring multiplication. Here 'cohomology' is usually understood as
1451:
334:{\displaystyle H^{\bullet }(X;R)=\bigoplus _{k\in \mathbb {N} }H^{k}(X;R).}
361:
127:
48:
17:
60:
40:
233:{\displaystyle H^{k}(X;R)\times H^{\ell }(X;R)\to H^{k+\ell }(X;R).}
55:, but the ring structure is also present in other theories such as
488:
A numerical invariant derived from the cohomology ring is the
368:
serving as the degree. The cup product respects this grading.
1306:, the mod 2 cohomology ring of the cartesian product of
1352:
1316:
1269:
1193:
1155:
1052:
1014:
938:
900:
797:
759:
669:
631:
514:
388:
256:
139:
74:
Specifically, given a sequence of cohomology groups
1367:
1334:
1291:
1255:
1177:
1141:
1036:
1000:
922:
886:
781:
745:
653:
617:
477:
333:
232:
1449:
243:The cup product gives a multiplication on the
71:on cohomology rings, which is contravariant.
356:) into a ring. In fact, it is naturally an
1433:, Cambridge: Cambridge University Press,
1355:
1318:
1240:
1229:
1211:
1099:
1088:
1070:
985:
974:
956:
844:
833:
815:
724:
706:
687:
569:
551:
532:
297:
1426:
1407:
1335:{\displaystyle \mathbb {R} P^{\infty }}
1450:
1385:vanishes except for the degree 0 part.
13:
1327:
1220:
1195:
1054:
965:
940:
799:
696:
671:
516:
14:
1469:
1368:{\displaystyle \mathbb {F} _{2}}
1346:variables with coefficients in
1279:
1271:
1250:
1244:
1233:
1207:
1165:
1157:
1136:
1117:
1109:
1103:
1092:
1066:
1024:
1016:
995:
989:
978:
952:
910:
902:
881:
862:
854:
848:
837:
811:
769:
761:
740:
734:
716:
683:
641:
633:
612:
593:
585:
579:
561:
528:
469:
443:
431:
421:
415:
389:
325:
313:
279:
267:
224:
212:
193:
190:
178:
162:
150:
1:
1401:
364:with the nonnegative integer
496:has cup-length equal to its
7:
1389:
1292:{\displaystyle |\alpha |=4}
1178:{\displaystyle |\alpha |=4}
1037:{\displaystyle |\alpha |=2}
923:{\displaystyle |\alpha |=2}
782:{\displaystyle |\alpha |=1}
654:{\displaystyle |\alpha |=1}
503:
10:
1474:
1410:Topology I, General Survey
344:This multiplication turns
247:of the cohomology groups
1342:is a polynomial ring in
494:complex projective space
67:of spaces one obtains a
1427:Hatcher, Allen (2002),
1408:Novikov, S. P. (1996).
1381:The cohomology ring of
371:The cohomology ring is
130:, which takes the form
90:with coefficients in a
1369:
1336:
1293:
1257:
1179:
1143:
1038:
1002:
924:
888:
783:
747:
655:
619:
479:
335:
234:
1370:
1337:
1294:
1258:
1180:
1144:
1039:
1003:
925:
889:
784:
748:
656:
620:
480:
336:
235:
126:) one can define the
1350:
1314:
1267:
1191:
1153:
1050:
1012:
936:
898:
795:
757:
667:
629:
512:
386:
254:
137:
1412:. Springer-Verlag.
53:singular cohomology
1430:Algebraic Topology
1396:Quantum cohomology
1365:
1332:
1289:
1253:
1175:
1139:
1034:
998:
920:
884:
779:
743:
651:
615:
475:
373:graded-commutative
331:
302:
230:
65:continuous mapping
57:de Rham cohomology
47:together with the
22:algebraic topology
498:complex dimension
285:
69:ring homomorphism
30:topological space
1465:
1443:
1423:
1374:
1372:
1371:
1366:
1364:
1363:
1358:
1341:
1339:
1338:
1333:
1331:
1330:
1321:
1298:
1296:
1295:
1290:
1282:
1274:
1262:
1260:
1259:
1254:
1243:
1232:
1224:
1223:
1214:
1203:
1202:
1184:
1182:
1181:
1176:
1168:
1160:
1148:
1146:
1145:
1140:
1135:
1134:
1116:
1102:
1091:
1083:
1082:
1073:
1062:
1061:
1043:
1041:
1040:
1035:
1027:
1019:
1007:
1005:
1004:
999:
988:
977:
969:
968:
959:
948:
947:
929:
927:
926:
921:
913:
905:
893:
891:
890:
885:
880:
879:
861:
847:
836:
828:
827:
818:
807:
806:
788:
786:
785:
780:
772:
764:
752:
750:
749:
744:
733:
732:
727:
715:
714:
709:
700:
699:
690:
679:
678:
660:
658:
657:
652:
644:
636:
624:
622:
621:
616:
611:
610:
592:
578:
577:
572:
560:
559:
554:
545:
544:
535:
524:
523:
484:
482:
481:
476:
468:
467:
455:
454:
442:
441:
414:
413:
401:
400:
340:
338:
337:
332:
312:
311:
301:
300:
266:
265:
239:
237:
236:
231:
211:
210:
177:
176:
149:
148:
92:commutative ring
39:formed from the
1473:
1472:
1468:
1467:
1466:
1464:
1463:
1462:
1458:Homology theory
1448:
1447:
1441:
1420:
1404:
1392:
1359:
1354:
1353:
1351:
1348:
1347:
1326:
1322:
1317:
1315:
1312:
1311:
1304:Künneth formula
1278:
1270:
1268:
1265:
1264:
1239:
1228:
1219:
1215:
1210:
1198:
1194:
1192:
1189:
1188:
1164:
1156:
1154:
1151:
1150:
1124:
1120:
1112:
1098:
1087:
1078:
1074:
1069:
1057:
1053:
1051:
1048:
1047:
1023:
1015:
1013:
1010:
1009:
984:
973:
964:
960:
955:
943:
939:
937:
934:
933:
909:
901:
899:
896:
895:
869:
865:
857:
843:
832:
823:
819:
814:
802:
798:
796:
793:
792:
768:
760:
758:
755:
754:
728:
723:
722:
710:
705:
704:
695:
691:
686:
674:
670:
668:
665:
664:
640:
632:
630:
627:
626:
600:
596:
588:
573:
568:
567:
555:
550:
549:
540:
536:
531:
519:
515:
513:
510:
509:
506:
463:
459:
450:
446:
434:
430:
409:
405:
396:
392:
387:
384:
383:
379:and ℓ; we have
307:
303:
296:
289:
261:
257:
255:
252:
251:
200:
196:
172:
168:
144:
140:
138:
135:
134:
109:
26:cohomology ring
20:, specifically
12:
11:
5:
1471:
1461:
1460:
1446:
1445:
1439:
1424:
1418:
1403:
1400:
1399:
1398:
1391:
1388:
1387:
1386:
1379:
1376:
1362:
1357:
1329:
1325:
1320:
1300:
1288:
1285:
1281:
1277:
1273:
1252:
1249:
1246:
1242:
1238:
1235:
1231:
1227:
1222:
1218:
1213:
1209:
1206:
1201:
1197:
1186:
1174:
1171:
1167:
1163:
1159:
1138:
1133:
1130:
1127:
1123:
1119:
1115:
1111:
1108:
1105:
1101:
1097:
1094:
1090:
1086:
1081:
1077:
1072:
1068:
1065:
1060:
1056:
1045:
1033:
1030:
1026:
1022:
1018:
997:
994:
991:
987:
983:
980:
976:
972:
967:
963:
958:
954:
951:
946:
942:
931:
919:
916:
912:
908:
904:
883:
878:
875:
872:
868:
864:
860:
856:
853:
850:
846:
842:
839:
835:
831:
826:
822:
817:
813:
810:
805:
801:
790:
778:
775:
771:
767:
763:
742:
739:
736:
731:
726:
721:
718:
713:
708:
703:
698:
694:
689:
685:
682:
677:
673:
662:
650:
647:
643:
639:
635:
614:
609:
606:
603:
599:
595:
591:
587:
584:
581:
576:
571:
566:
563:
558:
553:
548:
543:
539:
534:
530:
527:
522:
518:
505:
502:
486:
485:
474:
471:
466:
462:
458:
453:
449:
445:
440:
437:
433:
429:
426:
423:
420:
417:
412:
408:
404:
399:
395:
391:
342:
341:
330:
327:
324:
321:
318:
315:
310:
306:
299:
295:
292:
288:
284:
281:
278:
275:
272:
269:
264:
260:
241:
240:
229:
226:
223:
220:
217:
214:
209:
206:
203:
199:
195:
192:
189:
186:
183:
180:
175:
171:
167:
164:
161:
158:
155:
152:
147:
143:
105:
9:
6:
4:
3:
2:
1470:
1459:
1456:
1455:
1453:
1442:
1440:0-521-79540-0
1436:
1432:
1431:
1425:
1421:
1419:7-03-016673-6
1415:
1411:
1406:
1405:
1397:
1394:
1393:
1384:
1380:
1377:
1360:
1345:
1323:
1309:
1305:
1301:
1286:
1283:
1275:
1247:
1236:
1225:
1216:
1204:
1199:
1187:
1172:
1169:
1161:
1131:
1128:
1125:
1121:
1113:
1106:
1095:
1084:
1079:
1075:
1063:
1058:
1046:
1031:
1028:
1020:
992:
981:
970:
961:
949:
944:
932:
917:
914:
906:
876:
873:
870:
866:
858:
851:
840:
829:
824:
820:
808:
803:
791:
776:
773:
765:
737:
729:
719:
711:
701:
692:
680:
675:
663:
648:
645:
637:
607:
604:
601:
597:
589:
582:
574:
564:
556:
546:
541:
537:
525:
520:
508:
507:
501:
499:
495:
491:
472:
464:
460:
456:
451:
447:
438:
435:
427:
424:
418:
410:
406:
402:
397:
393:
382:
381:
380:
378:
374:
369:
367:
363:
359:
355:
351:
347:
328:
322:
319:
316:
308:
304:
293:
290:
286:
282:
276:
273:
270:
262:
258:
250:
249:
248:
246:
227:
221:
218:
215:
207:
204:
201:
197:
187:
184:
181:
173:
169:
165:
159:
156:
153:
145:
141:
133:
132:
131:
129:
125:
121:
117:
113:
108:
104:
100:
96:
93:
89:
85:
81:
77:
72:
70:
66:
62:
59:. It is also
58:
54:
50:
46:
42:
38:
34:
31:
27:
23:
19:
1429:
1409:
1343:
1307:
489:
487:
376:
370:
365:
357:
353:
349:
345:
343:
242:
123:
119:
115:
111:
106:
102:
98:
94:
87:
83:
79:
75:
73:
44:
32:
25:
15:
1383:suspensions
362:graded ring
128:cup product
97:(typically
49:cup product
18:mathematics
1402:References
1310:copies of
490:cup-length
245:direct sum
61:functorial
43:groups of
41:cohomology
1328:∞
1276:α
1248:α
1221:∞
1205:
1200:∗
1162:α
1122:α
1107:α
1064:
1059:∗
1021:α
993:α
966:∞
950:
945:∗
907:α
867:α
852:α
809:
804:∗
766:α
738:α
697:∞
681:
676:∗
638:α
598:α
583:α
526:
521:∗
461:α
457:⌣
452:ℓ
448:β
439:ℓ
425:−
411:ℓ
407:β
403:⌣
394:α
294:∈
287:⨁
263:∙
208:ℓ
194:→
174:ℓ
166:×
1452:Category
1390:See also
504:Examples
63:: for a
1302:By the
1437:
1416:
1263:where
1149:where
1008:where
894:where
753:where
625:where
24:, the
122:, or
86:) on
35:is a
28:of a
1435:ISBN
1414:ISBN
37:ring
101:is
16:In
1454::
500:.
118:,
114:,
110:,
1444:.
1422:.
1375:.
1361:2
1356:F
1344:n
1324:P
1319:R
1308:n
1299:.
1287:4
1284:=
1280:|
1272:|
1251:]
1245:[
1241:Z
1237:=
1234:)
1230:Z
1226:;
1217:P
1212:H
1208:(
1196:H
1185:.
1173:4
1170:=
1166:|
1158:|
1137:)
1132:1
1129:+
1126:n
1118:(
1114:/
1110:]
1104:[
1100:Z
1096:=
1093:)
1089:Z
1085:;
1080:n
1076:P
1071:H
1067:(
1055:H
1044:.
1032:2
1029:=
1025:|
1017:|
996:]
990:[
986:Z
982:=
979:)
975:Z
971:;
962:P
957:C
953:(
941:H
930:.
918:2
915:=
911:|
903:|
882:)
877:1
874:+
871:n
863:(
859:/
855:]
849:[
845:Z
841:=
838:)
834:Z
830:;
825:n
821:P
816:C
812:(
800:H
789:.
777:1
774:=
770:|
762:|
741:]
735:[
730:2
725:F
720:=
717:)
712:2
707:F
702:;
693:P
688:R
684:(
672:H
661:.
649:1
646:=
642:|
634:|
613:)
608:1
605:+
602:n
594:(
590:/
586:]
580:[
575:2
570:F
565:=
562:)
557:2
552:F
547:;
542:n
538:P
533:R
529:(
517:H
473:.
470:)
465:k
444:(
436:k
432:)
428:1
422:(
419:=
416:)
398:k
390:(
377:k
366:k
360:-
358:N
354:R
352:;
350:X
348:(
346:H
329:.
326:)
323:R
320:;
317:X
314:(
309:k
305:H
298:N
291:k
283:=
280:)
277:R
274:;
271:X
268:(
259:H
228:.
225:)
222:R
219:;
216:X
213:(
205:+
202:k
198:H
191:)
188:R
185:;
182:X
179:(
170:H
163:)
160:R
157:;
154:X
151:(
146:k
142:H
124:C
120:R
116:Q
112:Z
107:n
103:Z
99:R
95:R
88:X
84:R
82:;
80:X
78:(
76:H
45:X
33:X
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