1114:
The definition of a cup product is dual (and so is analogous) to the above definition of the intersection form on homology of a manifold, but is more abstract. However, the definition of a cup product generalizes to complexes and topological manifolds. This is an advantage for mathematicians who
373:
590:
1028:
1377:
simply-connected 4-manifold with positive definite intersection form has the diagonal (scalar 1) intersection form. So
Freedman's classification implies there are many non-smoothable 4-manifolds, for example the
238:
666:
191:
1109:
1228:
453:
744:
808:
471:
884:
1260:
The definition using cup product has a simpler analogue modulo 2 (which works for non-orientable manifolds). Of course one does not have this in de Rham cohomology.
1251:
1148:
922:
842:
1312:
1168:
230:
118:
938:
458:
This is well-defined because the intersection of a cycle and a boundary consists of an even number of points (by definition of a cycle and a boundary).
1339:
used the intersection form to classify simply-connected topological 4-manifolds. Given any unimodular symmetric bilinear form over the integers,
595:
Using the notion of transversality, one can state the following results (which constitute an equivalent definition of the intersection form).
1461:
1325:
The signature of the intersection form is an important invariant. A 4-manifold bounds a 5-manifold if and only if it has zero signature.
368:{\displaystyle \cap _{M,2}:H_{2}(M;\mathbb {Z} /2\mathbb {Z} )\times H_{2}(M;\mathbb {Z} /2\mathbb {Z} )\to \mathbb {Z} /2\mathbb {Z} }
1441:
602:
127:
1364:
465:
is oriented, analogously (i.e. counting intersections with signs) one defines the intersection form on the 2nd homology group
1363:
4-manifolds with the same intersection form are homeomorphic. In the odd case, the two manifolds are distinguished by their
1039:
1433:
1176:
384:
1359:
is odd, there are two, with at least one (possibly both) having no smooth structure. Thus two simply-connected closed
1484:
54:
of the 4-manifold. It reflects much of the topology of the 4-manifolds, including information on the existence of a
679:
1322:
smooth 4-manifold (or more generally one with no 2-torsion residing in the first homology), the converse holds.
757:
83:
1326:
585:{\displaystyle Q_{M}=\cap _{M}=\cdot _{M}:H_{2}(M;\mathbb {Z} )\times H_{2}(M;\mathbb {Z} )\to \mathbb {Z} .}
47:
1370:
1319:
932:
be a closed oriented 4-manifold (PL or smooth). Define the intersection form on the 2nd cohomology group
1479:
1330:
1401:
925:
896:
1119:
847:
1115:
are interested in complexes and topological manifolds (not only in PL and smooth manifolds).
1457:
1437:
17:
1236:
1133:
1023:{\displaystyle Q_{M}\colon H^{2}(M;\mathbb {Z} )\times H^{2}(M;\mathbb {Z} )\to \mathbb {Z} }
907:
821:
1336:
1282:
1254:
1153:
121:
55:
1451:
208:
96:
1269:
79:
71:
1276:
51:
1473:
1392:
1379:
902:
75:
31:
1419:
27:
A special symmetric bilinear form on the 2nd (co)homology group of a 4-manifold
1333:
implies that a smooth compact spin 4-manifold has signature a multiple of 16.
43:
1329:
implies that a spin 4-manifold has signature a multiple of eight. In fact,
39:
844:
has the sign +1 or −1 depending on the orientations, and
1170:, then the intersection form can be expressed by the integral
433:
810:
are represented by closed oriented surfaces (or 2-cycles)
668:
are represented by closed surfaces (or 2-cycles modulo 2)
661:{\displaystyle a,b\in H_{2}(M;\mathbb {Z} /2\mathbb {Z} )}
186:{\displaystyle a,b\in H_{2}(M;\mathbb {Z} /2\mathbb {Z} )}
1453:
Algebraic
Topology From Geometric Viewpoint (in Russian)
1412:
The topology of 4-manifolds, Lecture Notes in Math. 1374
818:
meeting transversely, then every intersection point in
232:, respectively. Define the intersection form modulo 2
1268:
Poincare duality states that the intersection form is
1104:{\displaystyle Q_{M}(a,b)=\langle a\smile b,\rangle .}
1285:
1239:
1179:
1156:
1136:
1042:
941:
910:
850:
824:
760:
682:
605:
474:
387:
241:
211:
130:
99:
1223:{\displaystyle Q(a,b)=\int _{M}\alpha \wedge \beta }
1279:4-manifold must have even intersection form, i.e.,
928:(and so an equivalent) definition as follows. Let
448:{\displaystyle a\cap _{M,2}b=|A\cap B|{\bmod {2}}.}
1306:
1245:
1222:
1162:
1142:
1103:
1022:
916:
878:
836:
802:
738:
660:
584:
447:
367:
224:
185:
112:
1343:, there is a simply-connected closed 4-manifold
1355:is even, there is only one such manifold. If
739:{\displaystyle a\cap _{M,2}b=|A\cap B|\mod 2.}
8:
1095:
1071:
803:{\displaystyle a,b\in H_{2}(M;\mathbb {Z} )}
201:modulo 2 viewed as unions of 2-simplices of
1284:
1238:
1205:
1178:
1155:
1135:
1047:
1041:
1016:
1015:
1005:
1004:
989:
975:
974:
959:
946:
940:
909:
855:
849:
823:
793:
792:
777:
759:
732:
731:
722:
708:
690:
681:
651:
650:
642:
638:
637:
622:
604:
575:
574:
564:
563:
548:
534:
533:
518:
505:
492:
479:
473:
436:
432:
427:
413:
395:
386:
361:
360:
352:
348:
347:
337:
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328:
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323:
308:
294:
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265:
246:
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216:
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176:
175:
167:
163:
162:
147:
129:
104:
98:
1118:When the 4-manifold is smooth, then in
7:
25:
1403:Intersection_number_of_immersions
727:
1301:
1289:
1195:
1183:
1092:
1086:
1065:
1053:
1012:
1009:
995:
979:
965:
873:
861:
797:
783:
723:
709:
655:
628:
571:
568:
554:
538:
524:
428:
414:
344:
341:
314:
298:
271:
180:
153:
18:Intersection form (4-manifold)
1:
1434:American Mathematical Society
1430:The wild world of 4-manifolds
676:meeting transversely, then
62:Definition using intersection
891:Definition using cup product
1450:Skopenkov, Arkadiy (2015),
1428:Scorpan, Alexandru (2005),
1264:Properties and applications
1130:are represented by 2-forms
1501:
1365:Kirby–Siebenmann invariant
894:
886:is the sum of these signs.
879:{\displaystyle Q_{M}(a,b)}
901:Using the notion of the
754:is oriented and classes
1347:with intersection form
1246:{\displaystyle \wedge }
1143:{\displaystyle \alpha }
917:{\displaystyle \smile }
837:{\displaystyle A\cap B}
48:symmetric bilinear form
1410:Kirby, Robion (1989),
1308:
1307:{\displaystyle Q(x,x)}
1247:
1224:
1164:
1163:{\displaystyle \beta }
1144:
1105:
1024:
918:
880:
838:
804:
740:
662:
586:
449:
369:
226:
187:
114:
1309:
1248:
1225:
1165:
1145:
1106:
1025:
919:
895:Further information:
881:
839:
805:
741:
663:
587:
450:
370:
227:
225:{\displaystyle T^{*}}
188:
124:. Represent classes
122:dual cell subdivision
115:
113:{\displaystyle T^{*}}
1327:Van der Blij's lemma
1283:
1237:
1177:
1154:
1134:
1040:
939:
908:
848:
822:
758:
680:
603:
472:
385:
239:
209:
128:
97:
1371:Donaldson's theorem
1275:By Wu's formula, a
897:Intersection theory
1485:Geometric topology
1314:is even for every
1304:
1243:
1220:
1160:
1140:
1120:de Rham cohomology
1101:
1020:
914:
876:
834:
800:
736:
658:
582:
445:
365:
222:
183:
110:
1463:978-5-4439-0293-7
1414:, Springer-Verlag
1394:Intersection form
1331:Rokhlin's theorem
1272:(up to torsion).
924:, one can give a
36:intersection form
16:(Redirected from
1492:
1466:
1446:
1424:
1415:
1406:
1397:
1337:Michael Freedman
1320:simply-connected
1313:
1311:
1310:
1305:
1252:
1250:
1249:
1244:
1229:
1227:
1226:
1221:
1210:
1209:
1169:
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1166:
1161:
1149:
1147:
1146:
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1110:
1108:
1107:
1102:
1052:
1051:
1029:
1027:
1026:
1021:
1019:
1008:
994:
993:
978:
964:
963:
951:
950:
923:
921:
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885:
883:
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877:
860:
859:
843:
841:
840:
835:
809:
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796:
782:
781:
745:
743:
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737:
726:
712:
701:
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665:
664:
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654:
646:
641:
627:
626:
591:
589:
588:
583:
578:
567:
553:
552:
537:
523:
522:
510:
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497:
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484:
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454:
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417:
406:
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374:
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366:
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332:
327:
313:
312:
297:
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270:
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257:
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231:
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166:
152:
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119:
117:
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111:
109:
108:
56:smooth structure
21:
1500:
1499:
1495:
1494:
1493:
1491:
1490:
1489:
1470:
1469:
1464:
1449:
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1427:
1418:
1409:
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1391:
1388:
1281:
1280:
1266:
1235:
1234:
1201:
1175:
1174:
1152:
1151:
1132:
1131:
1043:
1038:
1037:
1033:by the formula
985:
955:
942:
937:
936:
906:
905:
899:
893:
851:
846:
845:
820:
819:
773:
756:
755:
686:
678:
677:
618:
601:
600:
544:
514:
501:
488:
475:
470:
469:
391:
383:
382:
378:by the formula
304:
261:
242:
237:
236:
212:
207:
206:
143:
126:
125:
100:
95:
94:
64:
50:on the 2nd (co)
28:
23:
22:
15:
12:
11:
5:
1498:
1496:
1488:
1487:
1482:
1472:
1471:
1468:
1467:
1462:
1447:
1442:
1425:
1416:
1407:
1398:
1387:
1384:
1303:
1300:
1297:
1294:
1291:
1288:
1265:
1262:
1242:
1231:
1230:
1219:
1216:
1213:
1208:
1204:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1159:
1139:
1112:
1111:
1100:
1097:
1094:
1091:
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1082:
1079:
1076:
1073:
1070:
1067:
1064:
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1046:
1031:
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1014:
1011:
1007:
1003:
1000:
997:
992:
988:
984:
981:
977:
973:
970:
967:
962:
958:
954:
949:
945:
913:
892:
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888:
887:
875:
872:
869:
866:
863:
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854:
833:
830:
827:
799:
795:
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788:
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776:
772:
769:
766:
763:
747:
746:
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721:
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711:
707:
704:
699:
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689:
685:
657:
653:
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621:
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614:
611:
608:
593:
592:
581:
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570:
566:
562:
559:
556:
551:
547:
543:
540:
536:
532:
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521:
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508:
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487:
482:
478:
456:
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376:
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268:
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146:
142:
139:
136:
133:
107:
103:
63:
60:
52:homology group
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1497:
1486:
1483:
1481:
1478:
1477:
1475:
1465:
1459:
1455:
1454:
1448:
1445:
1443:0-8218-3749-4
1439:
1435:
1431:
1426:
1423:
1422:
1417:
1413:
1408:
1405:
1404:
1399:
1396:
1395:
1390:
1389:
1385:
1383:
1381:
1376:
1372:
1368:
1366:
1362:
1358:
1354:
1350:
1346:
1342:
1338:
1334:
1332:
1328:
1323:
1321:
1317:
1298:
1295:
1292:
1286:
1278:
1273:
1271:
1263:
1261:
1258:
1256:
1255:wedge product
1240:
1217:
1214:
1211:
1206:
1202:
1198:
1192:
1189:
1186:
1180:
1173:
1172:
1171:
1157:
1137:
1129:
1125:
1121:
1116:
1098:
1089:
1083:
1080:
1077:
1074:
1068:
1062:
1059:
1056:
1048:
1044:
1036:
1035:
1034:
1001:
998:
990:
986:
982:
971:
968:
960:
956:
952:
947:
943:
935:
934:
933:
931:
927:
911:
904:
898:
890:
870:
867:
864:
856:
852:
831:
828:
825:
817:
813:
789:
786:
778:
774:
770:
767:
764:
761:
753:
749:
748:
733:
728:
719:
716:
713:
705:
702:
697:
694:
691:
687:
683:
675:
671:
647:
643:
634:
631:
623:
619:
615:
612:
609:
606:
598:
597:
596:
579:
560:
557:
549:
545:
541:
530:
527:
519:
515:
511:
506:
502:
498:
493:
489:
485:
480:
476:
468:
467:
466:
464:
459:
442:
437:
424:
421:
418:
410:
407:
402:
399:
396:
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388:
381:
380:
379:
357:
353:
333:
329:
320:
317:
309:
305:
301:
290:
286:
277:
274:
266:
262:
258:
253:
250:
247:
243:
235:
234:
233:
217:
213:
204:
200:
196:
172:
168:
159:
156:
148:
144:
140:
137:
134:
131:
123:
105:
101:
93:. Denote by
92:
88:
85:
84:triangulation
81:
77:
73:
69:
61:
59:
57:
53:
49:
46:is a special
45:
41:
37:
33:
19:
1452:
1429:
1421:Linking_form
1420:
1411:
1402:
1393:
1374:
1369:
1360:
1356:
1352:
1348:
1344:
1340:
1335:
1324:
1315:
1274:
1267:
1259:
1232:
1127:
1123:
1117:
1113:
1032:
929:
900:
815:
811:
751:
673:
669:
594:
462:
460:
457:
377:
202:
198:
194:
193:by 2-cycles
90:
86:
74:4-manifold (
67:
65:
35:
29:
1480:4-manifolds
1380:E8 manifold
903:cup product
599:If classes
82:). Take a
32:mathematics
1474:Categories
1386:References
1270:unimodular
44:4-manifold
1456:, MCCME,
1373:states a
1318:. For a
1241:∧
1218:β
1215:∧
1212:α
1203:∫
1158:β
1138:α
1096:⟩
1078:⌣
1072:⟨
1013:→
983:×
953::
912:⌣
829:∩
771:∈
717:∩
688:∩
616:∈
572:→
542:×
503:⋅
490:∩
422:∩
393:∩
345:→
302:×
244:∩
218:∗
141:∈
106:∗
42:compact
40:oriented
1253:is the
205:and of
1460:
1440:
1375:smooth
1361:smooth
1351:. If
1233:where
80:smooth
72:closed
38:of an
34:, the
1122:, if
70:be a
1458:ISBN
1438:ISBN
1277:spin
1150:and
1126:and
926:dual
814:and
672:and
197:and
120:the
66:Let
750:If
729:mod
461:If
434:mod
89:of
78:or
30:In
1476::
1436:,
1432:,
1382:.
1367:.
1257:.
734:2.
76:PL
58:.
1357:Q
1353:Q
1349:Q
1345:M
1341:Q
1316:x
1302:)
1299:x
1296:,
1293:x
1290:(
1287:Q
1207:M
1199:=
1196:)
1193:b
1190:,
1187:a
1184:(
1181:Q
1128:b
1124:a
1099:.
1093:]
1090:M
1087:[
1084:,
1081:b
1075:a
1069:=
1066:)
1063:b
1060:,
1057:a
1054:(
1049:M
1045:Q
1017:Z
1010:)
1006:Z
1002:;
999:M
996:(
991:2
987:H
980:)
976:Z
972:;
969:M
966:(
961:2
957:H
948:M
944:Q
930:M
874:)
871:b
868:,
865:a
862:(
857:M
853:Q
832:B
826:A
816:B
812:A
798:)
794:Z
790:;
787:M
784:(
779:2
775:H
768:b
765:,
762:a
752:M
724:|
720:B
714:A
710:|
706:=
703:b
698:2
695:,
692:M
684:a
674:B
670:A
656:)
652:Z
648:2
644:/
639:Z
635:;
632:M
629:(
624:2
620:H
613:b
610:,
607:a
580:.
576:Z
569:)
565:Z
561:;
558:M
555:(
550:2
546:H
539:)
535:Z
531:;
528:M
525:(
520:2
516:H
512::
507:M
499:=
494:M
486:=
481:M
477:Q
463:M
443:.
438:2
429:|
425:B
419:A
415:|
411:=
408:b
403:2
400:,
397:M
389:a
362:Z
358:2
354:/
349:Z
342:)
338:Z
334:2
330:/
325:Z
321:;
318:M
315:(
310:2
306:H
299:)
295:Z
291:2
287:/
282:Z
278:;
275:M
272:(
267:2
263:H
259::
254:2
251:,
248:M
214:T
203:T
199:B
195:A
181:)
177:Z
173:2
169:/
164:Z
160:;
157:M
154:(
149:2
145:H
138:b
135:,
132:a
102:T
91:M
87:T
68:M
20:)
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